LIBRARY OF CONGRESS, 
©fjapHlj ©ommgfri !f 0........ 



UNITED STATES OF AMERICA. 



A COMPLETE MANUAL 



OX TEACHING 



Arithmetic. Algebra, and Geometry 



INCLUDING 



A Brief History of these Braxcees 



BY 



'ft* ■ 

/TO J. M. GREENWOOD, A.M., 



C/ Superintendent of Schools, Kansas City, Mo. 







NEW YORK : 
Effingham Maynard & Co., Publishers, 

771 Broadway and 67 & 69 Ninth Street. 






7* 



thomson's 
Mathematical Series. 



FIRST LESSONS IN ARITHMETIC. 

Oral and Written. Illustrated. 

•COMPLETE GRADED ARITHMETIC. 

Oral and Written. In one volume. 

Key to Complete Graded Arithmetic. 

(For Teachers only.) 

COMMERCIAL ARITHMETIC. 
Key to Commercial Arithmetic. 

{For Teachers only.) 



ILLUSTRATED TABLE-BOOK. 

NEW RUDIMENTS OF ARITHMETIC. 

COMPLETE INTELLECTUAL ARITHMETIC. 

NEW PRACTICAL ARITHMETIC. 

KEY TO NEW PRACTICAL ARITHMETIC. 

(For Teachers only.) 

NEW PRACTICAL ALGEBRA, 

KEY TO NEW PRACTICAL ALGEBRA. 
(For Teachers only.) 

COLLEGIATE ALGEBRA. 

KEY TO COLLEGIATE ALGEBRA. 
[For 7 cache rs only.) 



Copyright, i8c/d, by Effingham Maynard & Co. 



PREFACE. 



The following treatise in its scope and methods is unlike 
any other work yet offered to the public in this country. 

Two distinct lines of thought are developed in the 
treatment of each branch — the historical phase on the one 
hand, and the scientific method of presentation on the 
other. It is believed that the historical features herein 
presented will be gladly welcomed by all teachers and stu- 
dents of Arithmetic, Algebra, and Geometry, interested in 
the origin, growth, and present boundaries of these sci- 
ences. In the preparation of the historical sketches the 
writings of Professor De Morgan, President Edward 
Brooks, Professors Eugene Rouche, and Ch. De Combe- 
rousse, and the English and American Encyclopaedies were 
consulted as the most reliable, accessible sources of infor- 
mation. There has not yet appeared a historian of the 
mathematical sciences; and what has been written is scat- 
tered through many volumes, very difficult of access even 
to the specialist, and entirely beyond the reach of the 
average reader. 

The methods of teaching each subject presented will 
commend themselves to the discriminating judgment of 
intelligent teachers everywhere. Each topic is discussed in 
detail, and the teacher is advised along what lines to w T ork, 
and how the work may be successfully accomplished. No 
points are left in obscurity. Important matters are duly 
emphasized. The language employed is simple, plain, di- 
rect, and positive. Non-essentials are kept in the back- 
ground. The teacher is expected to think, to feel, to act 

iii 



iv PREFACE. 

and then to inspire his pupils with the same thoughts, 
feelings, and actions. 

The solutions of the difficult Arithmetical and Algebraic 
problems will be very helpful to a large class of teachers 
and students who are desirous of mastering some of the 
intricacies of these exact sciences. 

The volume is submitted with the hope that it may 
stimulate others more competent to carry forward a work 
of which this is only the beginning. 

J. M. Greenwood. 

Kansas City, Mo., July, 1890. 



CONTENTS. 



ARITHMETIC. 

PAGE 

Historical Sketch, 9 

Preliminary Remarks, 21 

Primary Arithmetic 23 

Things to be observed in teaching Primary Arithmetic, . . 25 

Teaching Primary Arithmetic, 26 

First Year's Work, 27 

Beading and Writing Numbers, 27 

Addition and Subtraction, 28 

Multiplication and Division, 29 

Fractions, 30 

General Directions for the First Year, 32 

Second Year's Work, 32 

Addition and Subtraction, 33 

Combinations of the Digits, 35 

Reading and Writing Numbers, 39 

Exercises in Fractions, 41 

Decimals, 41 

Third Year's Work, 42 

Terms and Signs, 44 

Making Change, 44 

Fourth Year's Work, 46 

Percentage and Interest, 49 

Fifth Year's Work, 50 

Extent of the Year's Work, 53 

Mental Arithmetic, 55 

v 



vi CONTENTS. 

PAGE 

Sixth and Seventh Year's Work 57 

Commission and Brokerage 59 

Interest, 61 

Promissory Notes, 62 

Discount, 62 

Bank Discount, 62 

Insurance, 63 

Stocks, 64 

Taxes, 64 

Compound Interest and Foreign Exchange, . . . .65 

Ratio and Proportion, 65 

Square Root and Cube Root, 66 

Series, 68 

Mensuration, 69 

Miscellaneous Problems, . . . . . . . .70 

Outlines, 70 

Mental Arithmetic, ........ 70 

Advanced Arithmetic, . . . , . ■ . . .72 

Rapid Methods of Adding, 72 

Some Contractions in Multiplication, 74 

Difficult Problems, 77 



ALGEBRA. 

Brief History, 101 

( >n Teaching Algebra, 112 

Theorems, 118 

Greatest Common Divisor and Least Common Multiple, . . 122 

Fractions, 123 

Equations of the First Degree, 128 

Equations of One Unknown Quantity, 130 

Equations of Two or More Unknown Quantities, . . . 132 

Elimination, 132 

The Form of Solutions, 133 

Other Methods of Elimination, 133 

Evolution, 142 

Involution and Evolution, 144 

Radical Quantities, 149 

Radical Equations, 156 



CONTENTS. Vll 

PAGE 

Quadratic Equations, . . .157 

Ratio and Proportion, 1T1 

Series, 173 

Geometrical Progression, 174 

Harmonical Progression, 176 

Continuation of Series, 177 

Reversion of Series, 178 

The Differential Method, . 179 ' 

Binomial Theorem, . ISO 

Logarithms, 181 

Permutations and Combinations, 182 

Probabilities, 185 

General Theory of Equations, 186 

Coefficients and Roots, . 187 

Sturm's Theorem, 191 

Horner's Method, 191 

Loci of Equations, . , . 193 



GEOMETRY. 

Historical Sketch, 195 

Teaching Geometry, ; 203 

Primary Conceptions, 203 

Definitions, Explanations, etc., 207 

Rectilinear Figures, 213 

Definitions and General Principles, 213 

Perpendicular and Oblique Lines, 213 

Parallel Lines and Angles, 217 

Triangles, 220 

Quadrilaterals 225 

The Circle, 227 

Construction of Problems, 234 

Area and Equivalency, 236 

Proportionality and Similarity, 239 

Problems in Equivalent Areas, 241 

Regular Polygons — Measurement of the Circle, . . . 242 

Maxima and Minima of Plane Figures, 246 



vi 11 CONTENTS. 

PAGE 

Geometry of Space, 248 

I Vlinitions and General' Principles, 249 

Diedral Angles, 250 

Polyedral Angles, 252 

Polyedrons, 255 

Regular Polyedrons, 258 

Euler's Theorem, 260 

The Three Round Bodies 261 

Cylinder, 261 

Cone, 262 

Sphere 263 

Sperical Surfaces and Volumes, 264 

Polar Triangles, 267 

Demonstrating Formulas, . . . . . . . . 269 

Modern Geometry, . . 269 

Transversals, 269 

Harmonic Proportion 270 

Anharmonic Ratio, 271 

Pole and Polar to a Circle, 273 

Reciprocal Polars, 274 

Radical Axis, 274 

Centers of Similitude, 275 

Conclusion, 276 



How to Teach Mathematics. 



Arithmetic. 



Historical Sketch. 

The origin of Arithmetic is unknown, but the term is 
derived from the Greek word Arithmos, number. Differ- 
ent nations have been accredited with the invention of this 
science. For instance, Josephus says: "For whereas the 
Egyptians were formerly addicted to different customs, 
and despised one another's sacred and accustomed rites, 
and were very angry with one another on that account, 
Abram conferred with each of them, and confuting the 
reasons they made use of, every one for his own practices, 
he demonstrated that such reasonings were vain and void 
of truth; whereupon he was admired by them in those 
conferences as a very wise man and one of great sagacity 
when he discoursed on any subject he undertook; and this 
not only in understanding it, but in persuading other men 
also to assent to him. He communicated to them Arith- 
metic, and delivered to them the science of Astronomy; 
for, before Abram came into Egypt, they were unacquainted 
with those parts of learning; and that science came from 
the Chaldeans into Egypt, and from thence to the Greeks 
also." 

9 



1 ' I ARITHMETIC. 

All that can bo inferred from the foregoing extract is 
that Abram had some knowledge of this science, which 
either originated with the Hebrews or was derived by them 
from surrounding nations. 

For a time the science was supposed to have originated 
with the Egyptians; others again ascribed it to the Chal- 
deans; while others gave the credit of it to the Phoenicians 
because of their nautical skill. Gradually each of these 
claims has faded away, and modern investigation points 
unmistakably to its origin as having been in India. It 
must not be inferred, however, that other nations did not 
possess some methods of computation or of counting. 
Counting is certainly coeval with the race. Even "barter," 
which is the most primitive method of exchange, could not 
be carried on between individuals without some system of 
estimating values by " how many and how much" While 
probably each of the great nations of antiquity had some 
method of computation, yet the origin and progress of the 
science, which has now reached a high degree of perfection, 
belong to India. 

The science in its modern form became possible only 
through the present system of notation. Consequently the 
most important event in the history of this science was the 
invention of the denary system. Among the ancient na- 
tions which possessed the art of writing, it was natural to 
represent numbers by letters. 

This we see from the Koman method of notation, which 
is a step in advance of the methods employed by the 
Greeks and the Hebrews. Both these nations used the first 
letters of their alphabets to represent numbers from 1 to 
10, with the exception that the Greeks inserted anew char- 
r to represent " G" so as to conform to the Hebrew 
notation, since there is no letter in the Greek language cor- 
>on<liiiL r to the sixth letter in the Hebrew. 

These two systems correspond "closely, character for 



HISTORICAL SKETCH. 11 

character, up to 80 f yet the Greeks had another notation 
for inscriptions which resembles the Eoman system quite 
closely. The Eoman notation with which we are familiar 
employs fewer characters than the Greek and admits of 
more and simpler combinations. Yet it is a clumsy system 
to work with, notwithstanding its advantages over the nota- 
tions used by the Hebrews and Greeks. 

The Arabic characters have been traced back to the 
Hindus, who, in turn, claim a divine origin for the inven- 
tion. As early as the fifth century of the Christian era 
the nine digits and zero, very nearly in the same form as 
we now have them, were known to the people of India, 
and not then as a recent invention, but as holding a per- 
manent position in their literature. Dr. Edward Brooks, in 
speaking upon this subject, says: " Among the sacred writ- 
ings of the Hindus there is preserved a treatise upon 
Arithmetic and Mensuration, written in the Sanscrit lan- 
guage, called Liliwati. This was regarded as of such 
inestimable value as to be ascribed by them to the im- 
mediate inspiration of Heaven. After an introductory 
preamble and colloquy of the gods, it begins with the expres- 
sion of numbers by the nine digits and the cipher, or small 
o. The characters are similar to those in present use, and 
the method of notation is the same. It contains the com- 
mon rules of Arithmetic and the extraction of the square 
root as far as two places. The examples are generally very 
easy, scarcely forming any part of the text, and are written 
in the margin with red ink. This work is very old, and 
proves that the Hindus have possessed this system for many 
centuries. Their knowledge of the science, however, is 
quite limited. They have no idea of the decimal scale 
descending, and their management of fractions is tedious 
and embarrassed." 

The general belief, till the discoveries made in Indian 
Literature, was that the present system of Arithmetic origi- 



13 ARITHMETIC. 

listed with the Arabs; but the honor of introducing it into 
Europe belongs to them. It appears that the Arabs ob- 
tained a knowledge of Arithmetic either directly from India 
during the seventh or eighth century, or from the Persians 
who had received it from the people of India. 

As early as the ninth century the Indian system of nota- 
tion was known to the Arabs. During the succeeding 
century it was in common use, at about which date it is 
Bupposed to have been introduced into Spain by the con- 
querors and thence spread gradually to the other European 
countries. However, the sources of information are so 
conflicting that it cannot be definitely determined at what 
date it was actually made known in Europe. Some author- 
ities are inclined to place it in the later part of the eighth 
century or early in the ninth, while others think it at least 
two centuries later. 

Professor Benjamin Greenleaf says: " It is evident that 
our numeral characters and our method of computing by 
them were in use among the Arabians about the beginning 
of the eighth century, when they invaded Spain, and it is 
probable that a knowledge of them was communicated to 
the inhabitants of Spain, and gradually to those of the 
other European countries." 

In the Encyclopaedia Britannica, Ninth Edition, is this 
statement: "The method was known to the Arabians in 
the ninth century, and in the course of the tenth it seems to 
have come into general use among them, especially in their 
onomical tables and their writings. It was probably in 
the following century that the Arabs introduced the nota- 
tion into Spain; but in regard to this we have no explicit 
information, and different accounts are given of the earliest 
instances of the use of the system in Europe. On the one 
hand, it is alleged that the figures first occur in a transla- 
tion of Ptolemy, of the date 113G, while others maintain 
that they were introduced (about 1252) by means of the 



HISTORICAL SKETCH. 13 

celebrated astronomical tables published by and named 
from Alphonso the Wise. That their use was known in Italy 
at the commencement of the thirteenth century appears to 
be satisfactorily established, for there is no good reason to 
doubt the genuineness of the MS. writings of Leonardo 
of Pisa, copies of which have been found bearing dates of 
1202 and 1220. Numerous other instances are given of 
the early use of the nine figures and the cipher, especially 
by astronomers, and in calendars/ 7 (Vol. II. , p. 4G1.) 

While this account of the introduction of figures into 
Europe appears quite reasonable notwithstanding the dis- 
crepancies in regard to dates, yet recent investigations tend 
to show a somewhat different account of the matter. The 
counter-claim is that the symbols or characters had been 
brought into Europe before the Arabs invaded Spain. 

Daring the ninth century, it is claimed, the Arabs learned 
the Indian Arithmetic from a work that is still extant, and 
that this treatise is founded upon a former collection of 
works brought from India to Bagdad about the year 7 73. 
The Arab treatise was translated into Latin during the 
Middle Ages, and became known partially to some European 
scholars. Also, upon careful comparison, it is thought by 
some that the figures used by the Europeans during the 
Middle Ages agree very closely with those used by the 
Arabs in Spain and Northern Africa, while they differ 
materially from those used by the Arabs in the East. Upon 
this difference it has been conjectured that the system the 
Arabs took to Spain was not the one, so far as the form is 
concerned, as that which they received from India. Again, 
it is inferred that the Neo-Pythagoreans, who taught the 
Greeks and Eomans the art of " ciphering," had learned 
directly from the people of India, and that Boethius and 
his successor used these figures in their mathematical hand- 
books, and thus they found way by degrees into European 
schools. 



1J: ABITIIMETIG. 

Tli is introduces the characters into Europe from Alexan- 
dria. From cumulative evidence it seems quite probable 
that the Indian Arithmetic was introduced into Europe 
through two sources: one through Egypt, probably during 
the fifth or sixth century, and the other passing through 
Bagdad and from thence into Europe by the way of Spain, 
somewhat later. That the characters bore a general re- 
semblance, though differing in some respects, to my mind 
is one of the strongest presumptions in support of a com- 
mon origin. Exact uniformity was impossible before the 
art of printing was known. That once discovered, the 
shapes of the characters are fixed. 

At present the earliest writers on Arithmetic, so far as 
is known, were Greeks. Their writings abound chiefly in 
matters of speculation. 

Pythagoras nearly 2500 years ^go attached great impor- 
tance to numbers. To-day we follow his classification of 
numbers into Prime and Composite, Perfect and Imper- 
fect, Redundant and Defective, Solid, Triangular, Square, 
Cubical, and Pyramidal. 

Odd numbers he imagined masculine, and even numbers 
as feminine. 

Euclid treats of numbers in the seventh, eighth, ninth, 
and tenth books of his Elements of Geometry. He dis- 
cusses proportion, prime and composite numbers; but from 
the fact that these books are omitted from the Elements, 
ept in Dr. Banrow's edition, the contribution to the 
nee of Arithmetic is of little value. 

About a century after, Eratosthenes invented a way of 
separating prime numbers from composite numbers. Be 
ribed the Beries of odd numbers on parchment. Then 
he cut out siu-h numbers as he found to be composite. 
The parchment thus cut was called a sieve. The method is 
known in mathematical literature as " Eratosthenes' Sieve." 

Diophantug of Alexandria wrote somewhat extensively 



HISTORICAL SKETCH. 15 

upon the properties of numbers. He composed thirteen 
books or chapters upon this subject; but seven of them 
were lost or destroyed, so that the contents of six only 
are really known. Diophantus is, however, much better 
known as a writer on Algebra than on Arithmetic. 

Following Diophantus a century or more later was 
Boethius, somewhat distinguished as an author, whose pro- 
duction was the classic work in Europe during the Middle 
Ages, and was regarded as a model for writers to imitate as 
late as the fifteenth century. This book is simply a curi- 
osity, as compared with our modern treatises upon the sub- 
ject. Boethius gave no rules for computing by numbers, 
but confined himself to the discussion of the properties of 
numbers. 

A work in manuscript discovered in the library in Cairo, 
written by Avicenna, an Arab phvsican who lived at Bok- 
hara about the year 1100 a.d., is believed to be the first work 
that employed the Indian characters and the decimal 
system. 

This treatise contains the four fundamental rules, besides 
many curious and interesting properties of numbers. It is 
frequently referred to as the oldest text-book, excepting, of 
course, the Indian treatises. 

We now enter upon the printing period of books; and 
notwithstanding the discovery and propagation of this art, 
it is difficult to decide who was the author of the first 
printed treatise on Arithmetic. 

Dr. Peacock and others contend that Lucas di Borgo, an 
Italian monk, is the author of the first printed Arithmetic, 
called Summa di Arithmetical published in 1494. This is 
also said to be the first book that used the Arabic characters, 
but Professor De Morgan is of the opinion that this treatise 
was preceded by the works of Oalandri and Peter Borgo. 
He admits that the treatise by Lucas di Borgo was the first 
on Algebra, but that the treatise of Philip Calandri on 



10 ARITHMETIC. 

Arithmetic was published in 1491, three years before the 
Arithmetic of Lucas di Borgo. 

After the appearance of the first few printed Arith- 
metics, others appeared in rapid succession, compared to the 
long intervals between writers of earlier times. 

In L501 John lluswit wrote a small Arithmetic in the 
German language. It was published at Cologne. He 
proved the fundamental rules by casting out the nines. 

Thirteen years later John Kobel published a book at 
Augsburg. The Arabic figures are inserted, but not used, 
by this author. He employed counters and the Roman 
letters instead of figures. 

The next Arithmetic was published in Paris, in 1515, by 
Qaspar Lax. The author added nothing new in this vol- 
ume of about two hundred and fifty pages. 

From this time onward there are evidences of original 
work. The science was added to by one and then by 
another. 

For instance, in 1522, Bishop Toustall, in his "Art of 
Computation," professes to have read all the books which 
had been published, and he says there was hardly a nation 
that did not have such books. 

The next writer of note was John Schoner, who edited 
Regiomontanus's Arithmetic in 1534. Regiomontanus was 
a celebrated German astronomer whose proper name was 
Johann Miiller. He demonstrated that the number of 
figures in a cube number could not exceed three times the 
number of figures in the root. 

Jerome Cardan, an Italian physician, mathematician, and 
author, ''celebrated for his science, self-conceit, and absurd 
vagaries," was born at Pavia in 1501. At the age of 38 
lie published at Milan his "Fraction Arithmetical a 
work of curious significance. The properties of numbers 
were treated of according to the manner of his predeces- 
sors, and then he dealt somewhat extravagantly upon the 



HISTOBICAL SKETCH. 17 

supposed mystic properties of numbers in foretelling future 
events. These properties he inferred from the numbers 
that he found in the Scriptures. 

This curious discovery is not so much a matter of sur- 
prise when it is remembered that " he dealt much in astrol- 
ogy and was a professed adept in the magical arts." His 
great epoch, says Hall am, is in the science of Algebra. 

The first Arithmetic printed in English in 1543 was 
written by Eobert Recorde, an eminent British mathema- 
tician, born about the year 1500. This book was entitled 
"The Ground of Arts: Teaching the Work and Practice 
of Arithmetic." Recorders original book has been greatly 
changed by the editors of different editions. They inter- 
larded the text with their own observations, — so much so as 
to leave the reader in doubt as to what Recorde originally 
wrote. A copy of the original edition is in the Greenville 
Library of the British Museum, and from it Mr. Heppel 
states that "Recorde calls the unit figure a digit, and the 
other parts 'articles;' thus, 5437 is made up of the articles 
5000, 400, 30, and the digit 7. In the table of contents 
we find, with more descriptive writing than is necessary to 
quote here, the main subjects: Declaration of the Profit 
of Arithmetic; Numeration, where Recorde does not ven- 
ture beyond ten thousands of millions; Addition; Sub- 
traction ; Multiplication ; Division ; Reduction ; Progres- 
sions; the Golden Rule; the Backer Rule [inverse pro- 
portion] ; the Rule of Double Proportion; the use of Fel- 
lowship, both with time and without time; and lastly, in 
Recorders words : e Unto all these is added their proof/ " 

John Timbs, F. S. A., in speaking of Robert Recorde, uses 
the following language : " Here should be mentioned the 
founder of the school of English writers, that is to say, to 
any useful or sensible purpose, — Robert Recorde, the physi- 
cian, a man whose memory deserves, on several accounts, a 
mtich larger portion of fame than it has met with. He was 



18 ARITHMETIC. 

the first who wrote on Arithmetic, and the first who wrote 
on Geometry, in English; the first who introduced Algebra 
into England; the first who wrote on Astronomy and the 
doctrine of the sphere in England; and, probably, the first 
Englishman who adopted the system of Copernicus. 

"Reoorde was also the inventor of the present method 
of ext meting the square root; the inventor of the sign of 
equality; and the inventor of the method of extracting the 
square root of multinomial algebraic quantities. According 
to Wood, his family was Welsh, and he himself a Fellow 
of All Souls' College, Oxford, in 1531. He died in 1558 in 
the King's Bench Prison, where he was confined for debt. 
Some have said that he w r as physician to Edward VI. and 
Mary, to whom his books are mostly dedicated. They are 
all written in dialogue between master and scholar, in the 
rude English of the time." ("School Days of Eminent 
Men/' page 118.) 

Three inventions are ascribed to Michael Stifel, a Luth- 
eran minister who published his Arithmetica Integra at 
Nuremberg in 1544. He used the signs +, — , and |/. 
Stifel acknowledges his obligations to Adam Eisca and 
Christopher Rudolph. 

Nicolas Tartaglia (" Stammer"), whose family name is 
unknown, published a mammoth treatise on Arithmetic 
and Algebra in the year 1556. It would require a volume 
to describe it, according to Professor De Morgan. 

Humphrey Baker, an English mathematician, published in 
L562 "The Well-Spring of Science," a very popular work 
on Arithmetic. 

"Of all works on Arithmetic prior to the publication of 

Crocker's celebrated book on the same subject (1668), this 

approaches nearest to the masterpiece of that 

celebrated arithmetician. ... It continued to be constantly 

reprinted till 1687, the latest edition we have met with." 

Biog. Diet.) 



HISTORICAL SKETCH. 19 

The author, so it appears, tried to bridge the chasm be- 
tween the pure abstract properties of numbers upon the 
one hand and the practical affairs of daily life upon the 
other. To adapt numbers to commerce and ordinary busi- 
ness was a wide departure, and marks an era in the 
science. 

Simon Stevinus, a Flemish engineer and mathematician, 
born at Bruges about 1550, made some very important dis- 
coveries in arithmetic, algebra, mechanics, and navigation. 
In 1585 he published at Leyden an Arithmetic, in which 
he devotes some space to Interest Tables and to Decimals. 
This is the first notice of Decimal Fractions. 

Albert Girard, another Dutch mathematician, edited 
Stevin's Arithmetic in 1634, and made such changes as he 
regarded as desirable. He changed the vinculum for the 
parenthesis. 

John Mellis revised Robert Recorders " Grounde of Arts," 
London, 1579, 1582,1590 (8vo); and in 1588 issued "Bookes 
of Accompts " (8vo). This is reputed to be the oldest Eng- 
lish work on Double-entry Book-keeping. 

In an old English work entitled "The Pathway of 
Knowledge," which was published" in London in 1596, are 
the following lines : 

" Thirtie daies hath September, 
Aprill, June, and November, 
Februarie, eight and twenty alone; 
All the rest thirtie and one. " 



» 



This book was translated from the Dutch by " W. P./ 
who is supposed to be the author of the quatrain. 

Pietro Antonio Cataldi, an Italian mathematician, was 
born at Bologna in 1548 and died in 1626. He was professor 
of mathematics in the university of his native city for more 
than forty years. 

At Bologna he founded an academy of mathematics which 



20 A1UTIIMETIC. 

is said to have been the most ancient institution of that 
kind known, but it was suppressed by the senate. 

Cataldi was an original investigator in different branches 
of mathematics. 

His discoveries in Arithmetic are methods of extracting 
the Bquare root of numbers, and the treatment of continued 
fractions. New ideas seem to have sprung up in his mind 
in great profusion, and he occupies a distinguished position 
among the Italian mathematicians of that century, and his 
works were used in more than a hundred towns and cities 
of Italy. 

The first English book containing tables of Compound 
Interest was a work by Eichard Witt, published in 1613. 
This book contained tables which the author called " Bre- 
viats," which were used to aid in the solution of problems 
in compound interest, annuities, rents, and so forth. 

The author appears to have used a line for the decimal 
point; that is, the tables were treated as numerators hav- 
ing 100 . . . for denominator, according to the number of 
figures in the numerator. 

Four years after the appearance of Witt's treatise John 
Napier published his arithmetic. He claims the invention 
of the decimal point, and he also states that Stevin (Stevinus) 
first discovered decimal fractions. Professor De Morgan 
is of the opinion that Napier did not invent the decimal 
point, but that he borrowed it from some other author. 

Robert Flood, an English physician and writer, born at 
Milgate in 1574, used P and M with strokes drawn through 
them. The first was the sign for addition; the second,for 
subtraction. The work containing these signs was pub- 
lished in 1617-19. 

A work written ten years later by Albert Girard does not 
use the decimal point. His method of expressing 23.375 
would be thus: 23/375. This indicates a carefulness in 
adopting new inventions. 



PRELIMINARY REMARKS. 21 

William Oughtred, an eminent divine and a distin- 
guished mathematician, born in 1573, introduced the sign 
X (St. Andrew's cross) in his Clavis Mathematical in 1631. 
Benjamin Martin says of Oughtred: "His style was 
very concise, obscure, and dry, and his rules and precepts 
so involved in symbols and abbreviations as rendered 
his mathematical writings very difficult to be understood.*' 

From that time to the present the science of Arithmetic 
has been perfected in many ways. More authors in Eng- 
land and in America have written on this subject since the 
days of Oughtred than on any other with the exception, 
perhaps, of English Grammar. 

Preliminary Remarks. 

The study of arithmetic should give clearness, activity, 
intensity, and tenacity to the mind on the disciplinary side; 
the drill or practical side should train to easy, quick, and 
accurate computation. 

Perception, attention, memory, imagination, judgment, 
and reason, are quickened and strengthened when the 
learner has grasped most firmly the fundamental principles 
of arithmetic, and he can apply them with just discrimi- 
nation to the solution of problems. The science of number 
requires the child to deal with things, relations, words, and 
thoughts. By close attention to these he becomes patient, 
logical, and systematic — habits of great value in the ordi- 
nary affairs of life. Self-mastery of principles is the only 
sure way to a clear understanding of this subject. Truth 
is many-sided. Some catch a glimpse from this side, others 
from that side, and so on. It is the living teacher whose 
presence, inspiration, directing power, can awaken thought 
and stimulate a class to its best and highest efforts. With- 
out soul-force, energy, and enthusiasm, — a love for truth 
and an overweening desire to search for it, to find it and 
retain it, — all education is naught. 



22 ARITHMETIC. 

While all true education in that higher sense is the gen- 
eralization of mental power and noble character, the 
science of arithmetic is peculiarly adapted to developing 
continuous and related thought, — in placing before the 
mind certain definite conditions from which must be de- 
duced necessitated conclusions. 

The child begins number concretely at first, but even 
then the memory and imagination run far ahead of "sense- 
products." To keep the child too long with "the sensuous 
and the known" is mental death. 

Right instruction in arithmetic requires that training 
which enables the learner to seize quickly the conditions of 
a question, and to hold them clearly and firmly and to 
examine them attentively till lie sees the conclusion. 

The advantages arising from a certain mechanical skill 
in obtaining results, which is frequently referred to as busi- 
ness or commercial arithmetic, should be secured in con- 
nection with the principles of the science rationally 
apprehended. The shop-keeping idea of arithmetic so 
prevalent among certain classes, while it may satisfy the 
superficial, is unworthy the name of the science which it 
belittles. To study is hard work. To concentrate, to 
direct, to prolong, to change effort, — to think closely, effec- 
tively, and successfully, distinguish the thinking man from 
the mere man. To solve problems is beneficial, but to 
solve problems and to think equally as well on other ques- 
tions is better still. 

To become a good arithmetician is the ideal that should 
be placed before every one who studies this science. The 
true teacher is the one who awakens and puts energy, en- 
thusiasm, activity, direction, and confidence into another 
mind and stimulates it to do its best. The very best work 
one can do always educates. Striving for something 
higher, nobler, grander, uplifts the soul and purifies 
the character. 



PBIMARY ARITHMETIC. 23 

The following are the essential conditions for teaching 
arithmetic: 1. A live, well-qualified teacher*, who under- 
stands child mind and knows how to teach one tiling at a 
time and how to teach that well. 2. A child that can be 
taught hoiv to sit or stand, how to study, how to think, koto 
to reason, an dhow to tell or write what he knows. 3. Books, 
slates, and pencils, blackboards, crayon, and erasers. 4. 
Apparatus for illustrations. 

Arithmetical Teaching will be presented under three 
subdivisions: 1. Primary. 2. Mental. 3. Practical and 
Higher. This classification is somewhat arbitrary in that 
Mental and Practical and Higher Arithmetic should be 
pursued simultaneously, and that the Mental in some form 
is connected with the Primary and Advanced. 

Primary Arithmetic. 

Counting is chiefly a matter of memory at first rather 
than a desire to find out the number of different objects 
in a group, or a collection, of things. In my own experi- 
ence I learned to count a hundred first, and afterwards I 
counted things to a hundred. Observations with children 
since confirm this view. This may not, however, be true 
of all children. Doubtless it is of a large majority. The 
fact that it exists indicates that tendency of the mind to be 
always reaching out in regions beyond the known. I hold 
it to be axiomatic that the child is working nearly all the 
time with things partially known. , The mind soon wearies 
with the known, and by virtue of its inherent energy is 
grappling with what is not yet its own. Complete compre- 
hension or thoroughness is not an attribute of childhood; 
but the child easily makes the transition from words to 
things, and from things to symbols of things. In count- 
ing, which is the first step in arithmetic, the word and the 
object may be developed together, or the child may learn 



J 1 ARITHMETIC. 

first to count, or call the words, "one, two, three," etc., 
without attaching any meaning to the words beyond the 
name. I am rather inclined to the opinion that the child 
when counting objects till the transition is fully made, will 
call each separate object by the name given it when it was 
counted. A little girl was counting for me, and an object 
was counted as "seven." If I took the object, which was 
an apple, and asked her how many, she replied "seven." 
So of other numbers. The name of the number was the 
name of the thing. A difficulty exists at this point, I im- 
agine, with all children. Counting, obviously, is the first 
thing for the child to learn. Till he can count and at- 
taches some idea to each number, progress in arithmetic 
is impossible. If the child is unable to count ten, I would 
practice him on counting till he learned it; then, by mem- 
or repetition, to a hundred. He soon "gets the hang 
of the thing," and it makes no material difference how he 
gets it, so he gets it. Some children reach a result one 
way, and some another. So he has a method that may be 
the best for him, but not for another whose eyes see in a 
different direction. 

Some minds are quick, but not retentive ; others slower, 
but retentive ; and a third class very slow, but very reten- 
tive. The best teacher always gives the dullest pupil a 
large chance. Nearly all depends, however, upon the man- 
in which this chance is offered. 

Returning to counting, all teachers do not follow the 
same line after the child can count. Personally, I want 
children to be able to count a hundred, and in teaching 
primary classes, I always found that they could do so 
with a little practice, if they had not already learned to 
count at home. Most children are taught to count before 
they go to school. Teach them if they do not know how. 
To count is the preliminary capital that the child must 
start with. Many teachers begin with objects. For some 



PRIMARY ARITHMETIC, 25 

children this is unnecessary. They perceive relations with- 
out the intervention of objects. Of recent years so many 
objects have been introduced at the wrong places in arith- 
metical instruction that great injury has been done to the 
learners, and their future progress retarded. The proper 
use of objects for purposes of illustration or of enforcing 
right ideas, cannot be overestimated. When a truth or a 
fact is thus once enforced and thoroughly apprehended, it 
is waste of time to continue illustrations longer. Just 
how long children should be kept playing arithmetic with 
objects must be answered by the intelligent teacher. 
There is perhaps more judgment to be used in knowing 
when to stop a thing of this kind than in beginning it. 
Enough is a feast, and too much is worse than a famine. 
As soon as a child perceives that the words "one, two, 
three," etc., are used in connection with objects it is time to 
lessen effort in this direction. 



Things to be Observed in Teaching Primary 
Arithmetic. 

1. The child must be taught to count things. 

2. The child must learn that words used in counting 
apply to objects as well as to numbers. 

3. Teach the child the figures and how to read them 
and how to make them. 

4. Teach facts by the use of objects till the child's under- 
standing is thoroughly reached. 

5. Let the child learn the difference between the spoken 
word, the written word, the figure or figures, and the 
letter or letters that represent a number. 

6. Teach one principle or fact at a time. 

7. Teach the child to do his best. 

8. Use small numbers to illustrate a principle. 

9. Encourage the child to make his own illustrations. 



516 ARITHMETIC. 

10. One illustration properly presented and understood, 
is better for the ordinary child than several different 
illustrations. 

11. Give the child a great deal of mental work. The 
more the better. 

12. Teach the child to think. 

13. Teach the child to tell concisely and connectedly what 
he knows. 

14. Teach the child to stand or to sit properly, and to 
breathe naturally. 

15. Let the instruction be of such a character as to 
cultivate the intellectual faculties and the will. 

Teaching Primary Arithmetic. 

1. Count objects and develop the idea of numbers, 
teacher and pupils to provide the objects to be counted. 
Sticks, books, marbles, pins, window-panes, seats, pupils, 
etc., can be used. 

2. Children must next be taught to write the figures 
that represent the objects counted. They should make 
the figures neatly and correctly on the slate, paper, and 
blackboard, and read them instantly. 

3. Let each class exercise on the board be made a slate 
exercise. 

4. When the pupils have learned to read the numbers 
from 1 to 10, or from 20 to 100, give them exercises to 
copy on their slates, at the next recitation. Little 
progress can be made in arithmetic till the learner is able 
to read and write numbers correctly and rapidly within 
certain limits. 

5. While the child is learning to read and to write 
numbers, he should be taught how to work with numbers. 

G. In his earlier efforts he should work with concrete 
exercises, each of which should be followed by the same 
changed into an abstract problem. 



BEADING AND WRITING NUMBERS. 27 

First Year's Work. 

Reading and Writing Numbers. 

I now assume that the learner can read and write 
numbers to 10, and that he can count to 100. Since he 
knows 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, it is an easy matter to 
teach him that 10 and 1 are 11, and so on to 20. That is, 
he learns all numbers from 10 to 20, and he can recall 
each at sight. From 20 go to 30 ; from 30 to 40 ; from 
40 to 50 ; and so on to 100. 

With the intermediate numbers the same plan is pur- 
sued, thus: 21 to 31, 31 to 41, 41 to 51, 51 to 61, 61 to 71, 
and so on. Sometimes the connection is more closely 
shown, thus: 2, 12, 22, 32, 42, 52, 62, 72, 82, 92, 102; or, 
5, 15, 25, 35, 45, 55, and so on. 

To teach quickness in reading numbers, the teacher can 
point rapidly to figures on the board, having one pupil 
answer while the others watch. It is better to point to 
several numbers in succession and have a pupil speak 
each number instantly. Avoid much concert answering. 
Skipping about is an excellent way to conduct such exer- 
cises. The child learns all small numbers just as he learns 
words — by their looks. 

It is really more trouble to teach children to read and 
write numbers from 1 to 100 than for them to learn to 
read and write from 100 to 1,000,000. After the child is 
quite familiar with numbers to 100, give him 200, 300, 
400, 500, 600, 700, 800 ? 900, 1000. Then he may after- 
wards take up 110, 210, 310, 410, 510, 610, 710, 810, 910; 
and 120, 220, 320, 420, 520, 620; and so on. 

There are many ways of reaching the same results. The 
differences consist in the time it takes some persons to get 
started, and then to go on after they start. They waste 
more time in getting ready than is required to teach the 
pupils the subject from the beginning. Any method that 



28 ARITHMETIC. 

is given here is simply suggestive. It is to be understood 
as a method and not as the method. A tree can be cut so 
as to fall in almost any direction; the same is true of 
chopping into arithmetical subjects. Much depends, 
however, upon the skill of the chopper. 

Addition and Subtraction. 

The ideas of putting two or more things or numbers 
into one group, and of separating a collection of things 
into two or more parts, are developed about the same time. 
Some go further and teach not only addition and sub- 
traction, but also multiplication and division, simultane- 
ously. 

Illustrative Exercises. 

1. One apple and one apple are how many apples ? 
Pupil to answer. 

2. One and one are how many? Pupil to answer. 

3. 1 -f- 1 = 2. Now is the time to show how plus is used. 

4. A boy had two apples and he lost one; how many 
apples had he left ? 

5. Two less one are how many? The pupil should now 
be taught minus and equal to. That is, 2 — 1 = 1. 

6. Give all such exercises to pupils orally first, and then 
written out afterwards. 

7. Two books and one more book are how many books? 

8. Two and one are how many? 
0. Two plus one are how many ? 

10. Write two plus one — equal to what? 

11. One, plus one, plus one, are how many ? 

12. Write this in arithmetical language. 

13. Mary picked three roses, and then gave one to Jane; 
how many roses were left ? 

14. Three minus one are how many ? 

15. Write three minus one — equal to what in figures? 



MULTIPLICATION AND DIVISION 29 

These and similar exercises may be continued by add- 
ing and then subtracting till 20 or even 30 is reached, 
or as far as the teacher may desire; first using 1, then 
2, 3, 4, 5, 6, 7, 8, 9, 10. Be sure that from use the pupils 
learn the nature of -\-, — , = , so that, whether the sign or 
its equivalent be used, there is no confusion. Arithmetic 
is a science of signs and symbols, and its language must be 
learned before much progress can be made. 

While working in these simplest of exercises, the teacher 
will push his pupils far beyond in reading and writing 
numbers. 

Many Primary Arithmetics furnish copious exercises. 

Multiplication and Division. 

Multiplication and Division rise naturally out of Addi- 
tion and Subtraction, and doubtless it is owing to this 
common origin that Griibe and his followers teach the 
four fundamental operations simultaneously. Instead, 
however, of teaching each operation as a separate and 
distinct process, the safer and, perhaps, better method is 
a compromise between the two. Too many changes in- 
troduce confusion, which may be avoided by taking one 
step safely and securely before attempting a second. The 
mind is so constituted that it passes readily from one 
process to its opposite with little effort, and this fact is 
significant as an educational principle. 

Exercises. 

1. John bought two apples at one cent each; what did 
he pay for both ? 

2. Two times one are how many ? 

3. 2x1 = what ? Let the pupil learn the name and 
the use of X . 

4. 1 X 2 = ?. 



30 ARITHMETIC. 

5. If an apple cost a cent, how many apples will two 
cents buy ? 

(>. One is contained in two how many times? 

;. 2 -s- 1 = what ? Then, 2-5-1 = 2. Teach the name 
and use of -K 

By gradual steps the teacher will lead his class to other 
concrete problems, followed by abstract ones, using all the 
smaller numbers first, both as multipliers and divisors. 

Thus far the facts of addition, subtraction, multipli- 
cation, and division have been partially developed with 
enough hints to enable the Primary teacher with the aid of 
any good elementary Arithmetic to start the pupil in the 
right direction. The theory of work should now be that 
of many oral and written exercises, involving all four pro- 

ses. Of course division necessitates the idea of Frac- 
tional numbers, which now require attention. 

Fractions. 

To teach the child to read fractional numbers, especially 
the simplest forms, is attended with no more difficulty than 
teaching him to read and write integers. He has the idea 
of a half, a third, and a quarter long before he enters 
school. They are words to him denoting things which have 
a real existence in form. For him to pass from these forms 
to |, J, i, \, f, f , etc., is an easy transition. As an evi- 
dence of this transition, the child knows quite well what is 
meant by the words half a biscuit, half an apple, half a 
pie, half a slate, and so of other fractional expressions. 
All that he really needs to learn is how to read and write 
them. 

In teaching them, I have pursued the following plan 
quite racoesrfully, teaching 1, H, \ % |, S, S, ?,, §, etc., till 
the class can read and write halves instantly. Then con- 
nect halves with multiplication and division ; as: 2 X i= ? 



FRACTIONS. 31 

2x1-? 2x2i=? 2X3=? 1X2 = ? lXH = ? 
1 X 3 = ? 1 X H .== ? Half of 5 = ? 2f X 2 = ? 

Or use concrete exercises, and then the abstract ones. 
After halves, which may be carried to any reasonable ex- 
tent, the idea of thirds is introduced and developed ; and 
fourths, fifths, and sixths follow naturally. The main point 
is, however, to carry multiplication and division by frac- 
tions along with the four fundamental rules till the pupil 
or class can add and subtract, multiply and divide, frac- 
tions as well as integers. 

By way of variety the pupil will add and subtract similar 
fractions, if he is started first with halves, as easily as he 
does integers. Any teacher can make this experiment and 
satisfy himself if he be skeptical. Let me urge again that 
the teacher mix concrete and abstract examples about 
equally, and he may use larger numbers as the pupils' 
faculties expand and are prepared to grasp larger and more 
complex conditions. 

So far the object has been to put the child during his 
first year in school in the way of using numbers intelli- 
gently, — laying, as it were, a foundation for him to build 
upon in future. 

It is no more difficult for the child to tell that the half of 
5 is 2^ than it is for him to say that " 3 " is half of " 6 "; 
or that "twice ten is twentv," and that the "half of 
twenty is ten." Doubling up, cutting " in two," are simple 
processes. The only caution — take short steps, and plan 
the child's work systematically. 

There should not only be a plan or method of teaching a 
child to read, write, add, subtract, multiply, and divide 
numbers for the year, but a plan of how to teach each sub- 
ject at the proper time and in the right way. Random, 
unsystematic teaching can never be very successful. 



32 ARITHMETIC. 

General Directions for the First Year. 

1. Be sure that the pupil can count objects correctly. 
\\ That he knows the figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 
at Bight, and that he can make tliem correctly. 

3. Teach Addition, Subtraction, Multiplication, and Di- 
vision by using small numbers, integral and fractional. Let 
the pupils illustrate these processes by using objects first, 
and abstract numbers afterwards. 

4. Teach the pupils how to sit and how to stand, and 
how to put their work in proper form on blackboard, slate, 
and paper. 

5. Give mental exercises for the purpose of leading pupils 
to do abstract work. 

6. Drill daily in reading and writing numbers. 

7. Teach the signs -{-, — , X, -S-, =, and how to use 
them. 

8. Give many combination exercises during the latter 
portion of the school year. 

9. Teach the pupils to count United States money ; and 
how to measure with their rulers, and how to measure with 
gill and pint cups. 

10. How to read and write numbers to 1000 for mini- 
mum work. Eoman notation to L. 

Pupils have no use for a text-book. 

Older pupils will do all this work in one third of the time. 

Second Year's Work. 

This year's work should be carried forward without using 
a text-book. There is no need of supplying a child with 
an Arithmetic before he can read intelligently what the 
book contains. A manual or guide-book for the teacher to 
Bhape her work by is required in all good schools, and it 
prevents that dissipation of energy which characterizes 
erratic work. The second year, or that period which cor- 



SECOND YEARS WORK 33 

responds to it with older and more mature children, is the 
time for fixing deep and permanent number forms in the 
child's mind. There are number forms, just as there are 
word forms and drawing forms, which need to be learned 
and retained. This, too, is the period when the child 
acquires skill in accurate and rapid calculation, if an oppor- 
tunity is given. Little children try their powers of speed 
in running and playing, and mentally they have like desires 
to gratify. Whether they shall always move demurely and 
soberly, or be let loose occasionally to caper, is a question that 
the intelligent teacher will not be long in deciding. Pre- 
mature age and soberness do not well become a lively, 
frolicsome child. There is something so unnatural about 
it that the genuine teacher of childhood cannot be in sym- 
pathy with it. We should keep our children young as long 
as possible. The green places in Arithmetic will give a 
vigorous glow to the body as well as a warm activity to the 
mind when properly presented by a master. 

Addition and Subtraction. 

During the first year many teachers lay special stress 
upon Addition, and all the energy of the pupil is conse- 
quently expended upon this subject. As a matter of choice, 
delay till the second year, or until that stage in the child's 
development is reached which corresponds to this period, 
rather than to begin so early. The chief object now is to 
perfect the child in the work which he has already com- 
menced. For this purpose the teacher should be supplied 
with several first-class primary arithmetics from which to 
select problems. But in addition to these aids other work 
should be included. For instance, the eye and. the memory 
should be trained in the study of arithmetic, as well as the 
reasoning faculties. The eye must be trained to see quickly 
and accurately, the memory to hold facts, objects, and re- 



:ll ARITHMETIC '. 

lations tenaciously, while the higher faculties deduce con- 
clusions from the data given and retained. Children can 
be taught to add numbers with as much accuracy as they 
read the lessons in their Reading Books, instead of blun- 
dering over each column of figures two or three times. 
Correct teaching should eliminate all uncertainty. This 
can be done by teaching first all integral forms of the 
nine digits, and secondly, the combinations of the digits 
themselves. 

These should be learned as- number forms, and recog- 
nized instantly by the pupil. These forms are: 1 = 1, 
2 = 1 + 1, 3 = 2 + 1=1 + 1 + 1, 4 = 2 + 2 = 3 + 

1 = 1 + 2 + 1 = 1 + 1 + 1 + 1, 5 = 4 + 1 = 3 + 2 = 

2 + 2 + 1 = 1 + 1 + 1 + 1 + 1, 6 = 5 + 1 = 4 + 2 = 
3+3 = 2 + 2 + 2 = 1+2 + 2 + 1, 7 = 6 + 1 = 3 + 

4 = 3 + 3+1 = 5 + 2 = 4 + 2 + 1, 8 = 4 + 4 = 3 + 

5 = 6 + 2 = 4 + 3 + 1, 9 = 6 + 3 = 8 + 1 = 5 + 4 = 

3 + 3 + 3. 

As an illustration, "6" is "3 + 3," "4 + 2," "5 + 1," 
or any other combination which produces it. Instead of 
seeing "4 + 2" as two separate numbers in Addition, they 
must be seen as " 6," and the same is true of all the other 
digits. " 9/' for instance, is " 7 + 2," "8 + 1," « 5 + 4," 
"6 + 3," or any other combination whose sum is "9." A 
little judicious practice in this direction will enable a class 
of small children to see a number in a group of numbers 
instantly. Quick sight and sharp practice are necessary 
to accomplish this object. 

Such questions as the following will enable the teacher 
to get a clear idea of what is intended in the above. 

1. What two numbers make 5 ? What three numbers 
make 5 ? What four numbers make 5 ? Write all the 
numbers that make 5. 

2. What three numbers make 9 ? What five numbers ? 



ADDITION AND SUBTRACTION 35 

Six numbers ? Write 9 in all the ways that you can. How 
many ways ? 

These exercises can be multiplied at pleasure. 

Combinations of the Digits. 

As the above table includes all necessary combinations 
to 9, those above 9 will now be considered. 

5 6 

1. 5 = 5 + 5 = 6 + 4 = 10. 2. 5 = 5 + 6 = 6 + 5 = 
7 4 

11. 3. 3 = 7 + 3 = 3 + 7 = 10. 4 . 7 = 7 + 4 = 4 + 7 

= 11. 5. 7 = 7 + 5 = 5 + 7 = 12. 6. 7 = 7 + 6 = 6 + 
7 2 

7 = 13. 7. 7 = 7 + 7 = 14. 8. 8 = 8 + 2 = 2 + 8 = 10. 

3 4 

9. 8 = 8 + 3 = 3 + 8 = 11. 10. 8 = 8 + 4 = 4 + 8= 12. 

5 6 

11. 8 = 8 + 5 = 5 + 8 = 13. 12. 8 = 8 + 6 = 6 + 8 = 
7 8 

14. 13.8 = 8 + 7 = 7 + 8 = 15. 14.8 = 8 + 8 = 16. 
1 2 

15. 9 = 9 + 1 = 1 + 9 = 10. 16. 9 = 9 + 2 = 2 + 9 = 

3 4 

11. 17. 9 = 9 + 3 = 3 + 9 = 12. 18. 9 = 9 + 4 = 4 + 

5 6 

9 = 13. is. 9 = 9 + 5 = 5 + 9 = 14. 20. 9 = 9 + 6 = 

7 8 

6 + 9 = 15. 21. 9 = 9 + 7 = 7 + 9 = 16. 22. 9 = 9 + 
9 

8 = 8 + 9 = 17. 23. 9 = 9 + 9 = 18. 

Drill the class on these combinations till they can tell 
the sum of any two numbers instantly. The above table 
goes more into detail than is necessary, perhaps ; but " 18," 
for instance, must be seen in all the forms under which 
" two 9's " can possibly appear. To teach these forms the 



36 ARITHMETIC. 

teacher can put on the board all combinations, and then 
pointing to each in rapid succession, receive prompt, accu- 

7 9 
rate replies. When the child looks at 6, or 4, or any other 
combination that is "13, " he should see "13" instantly as 
one number without going through, the slow steps of addi- 
tion by seeing and thinking over each number separately. 
Ten minutes' drill each day for a month or two will give 
the average pupil complete mastery of these simple combi- 
nations. The time will be profitably spent. 

Another excellent exercise in Addition consists in calling 
a rlass of pupils in front of the blackboard, and then hav- 
ing all stand with their backs to the board while the teacher 
writes a column of figures on the board to be added. At 
a given signal all the pupils face the board for a few sec- 
onds, and at a second signal each writes the answer, or 
retains it mentally. No pupil should have a chance to add 
the column the second time. Begin with a few figures in 
a column at first, and thus the pupils gain confidence in 
themselves. The most rapid additions I have ever seen, 
with the exception of one or two "Lightning Calculators," 
were by little children in the first and second year's work 
in city schools. They were " double-geared mental lightning 
let loose " and — the best of it all — each one added correctly. 
Dreamy, slow drilling never accomplishes anything. Sharp, 
quick, active, intelligent work tells. 

As a variation in "seeing, retaining, and adding num- 
bers," the following is eminently practical : Call the class 
in front of the board, standing with faces from the board. 
The teacher now writes a single column of figures on the 
board, and at a given signal all the class turn with faces to 
the board for an instant, and then at a signal turn from it, 
and at another signal write or speak the sum. Children 
with a little practice daily will take in a column of eight or 
ten figures in one or two seconds, aud will give the correct 



COMBINATIONS OF DIGITS. 37 

result nine times out of ten. Children that have been well 
trained in "seeing and adding" will add a column that 
amounts to 80 or 100 in four seconds. This speed is fre- 
quently attained by " straight addition," that is, taking 
each figure separately without grouping. 

For several years I have made a special study of how 
ch ildren add. The artifices they adopt from necessity form 
an amusing bit of experience in school life. 

A description of one exercise will be given which occurred 
recently. There were more than fifty children in the room 
— -second grade. The teacher had drilled the children quite 
well ; that is, they added a column of several figures rather 
rapidly.. She believed they added without counting, be- 
cause she had told them not to count, make marks, or do 
anything of that kind. The children looked away while 
she wrote a column of figures on the board, and when the 
word was given they turned their faces to the board and 
began to add. The position I occupied gave me command 
of each face, and the movements of lips, eyes, and fingers 
betrayed the workings of each mind. More than half the 
pupils used "some dodge;'/ but the teacher had discov- 
ered nothing, and when I asked her if she was satisfied that 
her children added the figures directly, she replied, "Cer- 
tainly !" She was requested to stand where she could see 
every face while I gave the class a similar exercise. This 
she did eagerly, and noted the results. 

A few of the artifices are given. A boy had written on a 
card the digits from 1 to 9. 

If he wanted to add "7," say, he counted on his " key- 
board " to 7, always beginning with "1" and actually count- 
ing up to the required number. It was absolutely aston- 
ishing how rapidly he counted. He made good time. 

A second boy added by "twos" — his father had taught 
him, and he broke all digits above two into ttvos ; and a 
third child added by threes. 



38 ARITHMETIC. 

A little German girl, if required to add 8 + 9 + 7, would 
take 1 from 8, and 2 from 9, and then say " 21, 24." That 
is, she added by " sevens'' — the only case of that kind I 
ever found. 

Counting fingers, buttons, marks, etc., were also employed 
by some. Without exaggeration, I am inclined to the opin- 
ion that many children use some artifices in Addition. That 
they dp resort to aids which they either devise or pick up 
is an evidence of erroneous instruction in the beginning. 
To start right and to keep right is the only way not to lose 
time in school work as well as in other pursuits. 

Since the pupil should be able to tell the sum of any two 
digits instantly, the teacher should not relax vigilance in 
number forms, but continue the work systematically and 
vigorously. Each step should be so well planned in Addi- 
tion as to fix the child's attention by its beauty and its nov- 
elty. Without saying so, it teaches a law by its exactness. 
Thus: 1. 

1 11 21 31 41 

1 _1 1 1 1 

2' 12' 22' 32' 42' 

The right-hand digit in the sums is the same. 

2 12 22 32 42 

a • o X 1 X ! 1 f 

Agam ' 2 ' 8' 18' 23' 88' 43' eta 

In a similar manner use 3, 4, 5, 6, 7, 8, 9. 



To illustrate: — - , — . — - , — , -— , — - , etc 



And, 



G 
9 


16 
9 


20 
9 


36 
9 


46 
9 


56 
9 


15' 


25' 


35' 


45' 


55' 


65' 


8 
9 

t r/ ' 


18 
9 

nrj > 


28 
9 

on > 


38 
9 


98 
9 


etc. 



READING AND WRITING NUMBERS. 39 

The method is this : Add the digit to a series of num- 
bers which increases by ten till the child sees the law. It 
is teaching one thing at a time. 

After each row of additions call attention to the unit's 
figure, thus: 

9 19 29 39 49 59 69 
_9 9 _9 _9 _9 _9 _9 
18' 28' 38' 48' 58' 68' 78' C * 

The unit "§" is the important point. Fix it deep in the 
mind. This is designed to impress forms that must be ab- 
solutely remembered for all time. The teacher can mul- 
tiply exercises at pleasure. 

Reading and Writing Numbers. 

The processes of Reading and Writing Numbers should 
now be carried forward with energy. Practically there is 
no limit within reason that can be assigned as a fixed 
boundary on this point, and beyond which pupils cannot 
pass. Problems in addition, subtraction, multiplication, 
and. division should be continued. The exercises should 
be of such a character as to test the pupil's knowledge 
and skill — just severe enough to make him do his best. 
The teacher must avoid too simple exercises. They en- 
feeble the pupil's powers. One problem that makes a 
child think is worth more than a hundred easy problems. 
Not-work hurts children. They are made dull, ignorant, 
lifeless, when kept doing that which amounts to nothing. 
To wear children out, confine them, and then give them 
easy, silly things to do. 

In addition, subtraction, multiplication, and division, 
use larger abstract numbers usually, rather than concrete 
numbers. Abstract numbers should be employed to secure 
accuracy; concrete examples to develop the thinking facul- 



40 ARITHMETIC. 

In the explanations of concrete problems, pictorial 
diagrams ami other graphic methods of illustration should 
be employed by the pupils. All work of whatever character 
ought to be neatly done. 

During this year's work is the proper time to commence 
teaching the tables of length, weight, and measure. The 
foot rule with which all pupils are supplied, can be used to 
teach linear measure. The pupils ought to measure. the 
lengths of familiar objects in the school-room till they can 
do so with considerable exactness. Two or three recitations 
will be enough to show them how to measure ; or rather, 
the teacher should see that they do it well. Rulers having 
the metric system on one side and our common system on 
the other are to be preferred. But at first the children 
should use the common measure. Let the children esti- 
mate heights and distances by the eye. Avoirdupois 
w T eight is best taught with "the scales:" Let the children 
learn to weigh articles at the grocery-store if in no other 
way. A gill, a quart, and a gallon measure, with a 
bucket of water, are sufficient to teach "liquid measure." 
Here, too, the children should do the " dipping and pour- 
ing." 

From these suggestions the skillful teacher can devise 
other methods to aid in this work. 

The following will illustrate what was suggested in re- 
gard to the graphic representation of problems. 

Problem. There were two gallons of molasses in a tin 
can; a pint leaked out, and three quarts were used; how 
many pints remained in the can ? 

This problem may be represented pictorially. 1. A tin 
can drawn to a scale to represent two gallons. 2. A quart 
measure drawn to a scale to indicate that vessel. 3. A pint 
cup drawn to a scale. 4. Let the pupil show by drawings 
how many quarts, or how many pints, two gallons equal. 
5. Let him show by lines what part of two gallons one 



EXERCISES IN FRACTIONS-DECIMALS. 41 

pint is. 6. What part of two gallons three quarts are. 
7. What part of the two gallons is that remaining in the 
tin can 9 

Exercises in Fractions. 

This work should be continued from that of the preced- 
ing year, but it should take a wider scope, and much of 
the work ought to be oral rather than written. 

To illustrate : If a pound of ginger costs 9 cents, what 
will two pounds cost? What will a half-pound cost? 
What will a third of a pound cost? What will a fourth 
of a pound cost? A fifth of a pound? A sixth of a 
pound ? Etc. 

Again : If a boy can chop a cord of wood in 10 hours, 
how many hours will it take him to chop a half-cord? A 
fourth of a cord ? A cord and a third ? Five eighths of 
a chord? Etc. 

In addition to solving problems containing fractions, the 
pupils should have considerable practice in reading and 
writing fractions, as well as continue addition, subtraction, 
multiplication, and division. The problems should be 
selected with reference to teaching methods of work, and 
to test the pupils' skill in solving them. No problem should 
be given because it is easy, unless for the purpose of lead- 
ing to another or others that are more difficult. 

Children wall work with fractional numbers just as well 
as with integral numbers if they are put at it in the right 
way. There is no good reason why fractions should be 
postponed till the child has been in school three or four 
years. 

Decimals. 

Children during this year can be taught to read and to 
write decimals, as w r ell as to perform the simpler operations 



42 ARITHMETIC. 

in decimals. The first step is to show that 

1-1 *- - 2 - 3 - - 3 

10 ' io • ' 10 - • ' 

are more easily written decimally than as common fractions. 
The pupil is led to Addition and Subtraction as fol- 
lows : 

1 I 1 2 O 4. 

Io + Io = io = - 3 > etc ' 

3 2 1 
Subtraction : -- — — = — = .3 — .2 = .1. 

Multiplication and Division in a similar manner. 

Also, 4 T V = 4.1 ; 3 T V = 3.7 ; l T 3 o = 1.3 ; and so on. 

Again, 4 T V + 3 T V = 4.7 + 3.7 = 8.4. 

The above are sufficient to suggest how the work can be 
approached. 

The teacher will observe that the pupil is encouraged to 
solve the problems in a rational manner first, and then to 
invent a method of illustrating each exercise when it can 
be done. The teacher must not forget that he is laying a 
good, broad foundation for the pupil's future mathematical 
work. 

The terms, as such, employed in the fundamental rules 
ought to be learned by the pupils during the second 
year. Let nothing be taught that must be unlearned. 

Third Year's Work. 

This year's work is simply an enlarged continuation of 
what has been previously outlined. Abstract exercises in 
small numbers and also in large numbers should constitute 
a large part of the pupil's work this year. These exercises 
are designed to give accuracy and rapidity in computation. 
It is better, at recitation, having ascertained the pupils' 



THIRD YEARS WORK. 43 

speed of work, to time the class on each problem. The 
main point is to secure absolute accuracy at the first 
trial. Much, indeed very nearly all. depends upon quick, 
sharp, correct training at this stage of the learner's prog- 
ress. Do not be afraid of giving the children large 
numbers in addition, subtraction, multiplication, and di- 
vision to work with. In division, the exercises need to be 
more carefully graded than in the other rules. The oral 
work in addition may include such exercises as 23 + 43 
= p - 21 _i_ 72 — 34 = ?, to be added at sight. The pupil 
is to get at the problem in his own way. But the teacher 
should help him finally in solving it in the easiest man- 
ner. A little practice on such problems will be time well 
spent. The exercises need not be confined to addition. 
Some time during the year an Elementary Arithmetic is 
usually introduced for the pupils to study. Since pupils 
waste two thirds of their time in arithmetic in trying 
to understand just what is required in the problems, 
considerable time may be profitably occupied in having 
pupils at recitation read problems and explain them. When 
a child knows what is to be done, usually all doubt is 
removed, and this is really the important step in solving 
problems, and it is one reason why certain pupils succeed 
so much better than others in arithmetic. They interpret 
problems more understand ingly. The best way, perhaps, 
to find out this fact is to assign problems for the pupils to 
tell what conditions are given, and what required. The 
proper questions are : What does the problem tell ? What 
is to be found out ? How will you go to work to find it 
out ? The power of correctly interpreting what is written 
or printed is a matter of great importance in teaching. 
Problems given orally by the teacher usually have this ad- 
vantage over printed conditions — that of the teacher's voice, 
which states the conditions more clearly and forcibly, and 
the emphatic points are more easily perceived. The teacher 



•44 ARITHMETIC. 

vivifies the problem, sets it in a strong light before the 
children's minds, and they seize all its conditions readily, 
In expressed problems, such as 

864 X 15 - 3G1 -4- 19 + 324 - 15f X 4 = ?, 

the pupil must be taught to interpret the mathematical 
language correctly. To know whether the pupil under- 
stands what he reads, he should be called upon to read it, 
and to explain it. 

If the children are studying one text-book, the teacher 
should require them to read from other sources similar 
problems, and to tell what each problem means. 

Terms and Signs. 

The following terms need to be thoroughly impressed : 
Sum, Amount, Minuend, Subtrahend, Difference, Remain- 
der, Multiplicand, Multiplier, Product, Dividend, Divisor, 
Quotient, Fraction, Numerator, Denominator, Line, Sur- 
face, Area, Volume, United States Money, Measures of 
Length, Liquid Measure, Dry Measure, Avoirdupois JVeight, 
Tinie by Clocks and Watches. 

Making Change. 

The pupils have now reached that stage in Arithmetic 
when they should be able to make change, or to know, 
when they purchase articles, the amount of money the 
article or articles cost, and what they are to receive in 
return. The teacher will show the pupils how to make 
out an itemized statement of a Bill of Goods, and how to 
receipt it This is best illustrated by making out simple 
purchases at first. A purchase of a few things at a grocery- 
store, properly itemized, will serve as a model, namely: 
George Miller bought of Thomas Jones 3 pounds of coffee 
at 15 cents a pound, 6 pounds of sugar at 10 cents a pound, 



MAKING CHANGE. 45 

| bushel of potatoes at SO cents a bushel. He gave Thomas 
Jones a two-dollar bill in payment. What did the articles 
cost, and how much in change should Thomas Jones hand 
George Miller? 

Form. 

Kansas City, 

September 20, 1S89. 
George Miller, 

Bought of Thomas Jones. 
1889. 
Sep. 20. To 3 pounds of coffee at 15 c. a pound. .45 
" 20. " 6 pounds of sugar at 10 c. a pound. .60 
" 20. " ± bushel of potatoes at SO c. a bush., .-40 



$1.45 

Other exercises can be multiplied at pleasure from actual 
business transactions. The children should be encouraged 
to make problems of their own, and to put them into 
proper form. Let the work be done neatly. 

In the tables and their applications in Compound 
Xumbers the pupil should see and handle the various 
measures in order to get clearer and better conceptions of 
the terms used in measurement. To correct their ideas of 
size, distance, weight, volume, etc.. they "should guess" 
at objects, such as pitchers, buckets, boxes, barrels, etc., 
and then find out what each holds. 

This is really the measuring period in arithmetical 
studies, and the pupils need to make the most of it. 
Size, weight, height, distance, and volume of objects can 
be closely approximated after the pupils have had con- 
siderable experience in correcting their own judgment in 
such matters. 

Strengthen weak pupils in reading and writing numbers. 



46 ARITHMETIC. 

first in integers, and secondly in fractions — both com- 
mon and decimal. Much practice and little theory should 
be the motto now. Teach pupils to work problems well 
one way. Make each pupil strong in addition and sub- 
traction, and very strong in multiplication and long divi- 
sion. Let the work half the time be mental, judiciously 
mixed with written work. Devices, such as are found in 
Beveral elementary arithmetics for securing rapid com- 
putation, should be employed by the teachers. Be sure 
that the multiplication-table is known with unerring cer- 
tainty. 

What has been marked out as a course of study in 
Arithmetic for three years applies to young children 
entering and continuing in graded schools; but a child ten 
or twelve years old, or older, can do all this work and even 
more in one year. 

In fact, a great deal of the little work can be omitted 
entirely with older children. 

Fourth Year's Work. 

It is still necessary to review notation and numeration, 
and to impress the difference between the simple and local 
values of the digits. Some time should be given to reading 
and writing the Roman notation. The work in Bills 
and Accounts, carried out in extended statements, should 
be continued from the previous grade. This work will be 
furthered if the pupils, instead of fictitious persons, are 
made parties to the transactions. 

In this connection the following terms, Btiyer, Seller, 
Creditor, Debtor, Credit, Debit, Account, Balance, State- 
ment, Payment, Receipt, To, @, ought to be mastered by 
the pupil, and then embodied in the exercises. When 
the idea dawns upon the pupil's mind that his knowledge 
of arithmetic can be applied to business matters, he is 



FOURTH YEARS WORK. 47 

stimulated to greater effort because of the use he is able to 
make of what he already knows. But at no stage of the 
pupil's progress will it do to relax effort in the four 
fundamental rules. Abstract numbers are to be used 
chiefly to secure accuracy and expertness. Let the teacher 
bring out by contrast the differences as expressed by wliole 
numbers, denominate numbers, fractional numbers, and 
decimal numbers. By a little reflection the pupil learns 
that he can add, subtract, multiply, and divide each of 
these numbers. When working with a number, the pupil 
must keep in mind the kind and meaning of the number, 
and the relation it bears to other numbers of a higher or 
lower denomination. This involves the comparison of one 
number with others, which cannot be effected unless the 
idea of similarity or likeness be clearly comprehended. 

This is the period in the learner's work when he must 
be thoroughly drilled in fractions, factoring, cancellation, 
and reduction. Unless he reads and writes numbers 
readily, and is accurate and rapid in handling numbers in 
the four fundamental rules, he will make no headway in 
fractions. After the pupil is thoroughly familiar with the 
processes of reduction of fractions, and knows exactly how 
to make dissimilar fractions similar, so as to add, subtract, 
and divide them, he should compare simple addition, com- 
pound addition, and addition of fractions, and note par- 
ticularly how these processes agree with one another. In 
a similar manner, he should trace the comparison through 
subtraction, multiplication, and division. Let problems 
be selected to illustrate these several relations. As soon as 
a pupil finds out that he can hitch new knowledge to what 
he already has, he has made a rapid stride forward in his 
generalizations. Following in the line of work is the sub- 
ject of decimals. Here again let the analogies in the four 
fundamental operations in simple numbers and the corre- 
sponding ones in decimals be discussed. To show the con- 



48 ARITHMETIC. 

nection between common fractions and decimals, simple 

exorcises, as 

l = .l, i = .2, etc., and ^ = .01, ± = .05, 

etc., are to be used. 

The fact to be taught is this: Decimals are common 
fractions with the dividing line and denominator omitted. 
The decimal point stands for the line and the denominator 
of the common fraction. 

The two methods of multiplying and dividing fractions 
deserve special attention. Let the pupil make problems 
and illustrate the methods. Be sure that these operations 
are mastered. Whenever the teacher calls for a specific 
thing to be done in definite arithmetical language, the 
pupil should know exactly what is meant and how to 
proceed. Precision in language and in thought are of 
the utmost importance in all mathematical studies. If a 
good foundation is laid in the fundamental rules, factor- 
ing, cancellation, reduction, common and decimal frac- 
tions, and an intelligent application of these rules to the 
measurement of length, area, and volume of forms, the 
pupil is prepared to take up the simpler cases in per- 
centage. 

The necessity, however, of understanding decimals well 
before beginning percentage is so apparent that special 
attention is called to this subject. The language of decimal 
notation and numeration is positive, and must have a per- 
manent place in the learner's mind. All vagueness must 
disappear and positive, definite, sharp-cut knowledge take 
its place. On all points the teacher must k?iow that the 
learner knows how to write, read, add, subtract, multiply, 
and divide decimals with absolute accuracy. The placing 
of the decimal point properly is the important matter in 
this subject. If the pupil can read and write decimals, he 



PERCENTAGE AND INTEREST, 49 

will experience little difficulty in addition and subtraction. 
But in multiplication he must watch for the decimal 
places in multiplicand and multiplier and product; and in 
division, when the decimal places in the dividend exceed 
those in the divisor, or when equal or less, and the 
reverse, and the effect on the quotient under all possible 
conditions. In short, the pupil must know exactly ivliere 
to put the decimal point in every operation. 

Percentage and Interest. 

The pupil adds new terms now to his arithmetical 
vocabulary, and it saves time to master the following at 
the outset: Per Cent ; Rate Per Cent; Sign of Per 
Cent (fc) ; Base; Percentage. If the pupil does not get 
the meaning from the definitions in his book, the teacher 
should make the terms plain to the pupil. The key to 
percentage is this: That 1 per cent of a number is T ^ 
part of that number, 2$ = T f ■$, ofo = T f F , etc. ; or, deci- 
mally, 1H = .01 = T fo, %$ = .02 = T f^ 3£ = .03 = T f ¥ , 
etc. Let the pupil express "per cent " in these three ways 
till he knows each well. 

Another point to dwell upon is that of fractional 
equivalents translated into " per cents," and the reverse. 

When 10fo is mentioned, the pupil should think along 
with that T V; 20fc, 4; 30$, y 3 ^; 5£, -fa; and so on. Fre- 
quently it is simpler to use the fractional equivalents, 
because they are smaller. 

To illustrate and impress the three cases of percentage, 
the greatest variety of problems ought to be used, and 
the pupils required to work them out under each of the 
three ways of expressing the notation. 

Under Interest, the "Six Per Cent Method," owing to 
its simplicity and universal application, is the one that 
should be taught first. When it is mastered the pupil 



50 ARITHMETIC. 

can find the interest at other rates by increasing or 
decreasing the interest at " (> ." 

It is better to teach this method well and have the 
pupil use it exclusively for a long time than to teach par- 
tially several different methods. To do a thing well in one 
way is Ear more valuable to the pupil than to half-way do it 
in half a dozen ways. 

In connection with Interest the pupils should be in- 
structed how to write notes and how to make credits. 
When the pupils are parties to the transactions a reality 
is given to the work that is very helpful. 

If a class is strong and up fully with its work, in 
addition to the topics already outlined in this grade the 
following may be touched upon more in detail, namely: 
1. Carpeting rooms. 2. Plastering and painting. 3. Board 
measure. 4. Stone and masonry. 5. Bins, wagon-boxes, 
tanks, and some knowledge of Discount as used in 
business. 

The work should always be neatly done, properly 
punctuated, and ready to be set up in type if necessary 
to do so. The pupil must now do more thinking and 
less "ciphering." Intelligent thought-work is the only 
work that pays. Rapid oral work induces thought. All 
dreamy, humdrum work should be avoided as a deadly 
mental poison. 

Fifth Year's Work. 

The preceding year's work took the pupil through what 
is usually called the "Elementary Arithmetic." In many 
graded schools the four years' course already outlined em- 
braces five years; but this is arithmetic long drawn out. 
Beginning now with a Practical or Common School Arith- 
metic, the pupil has arrived at that stage in the develop- 
ment of his reasoning faculties when he must do more 
thinking and less mechanical ciphering. He can now take 



FIFTH TEARS WORK. 51 

hold of principles and apply them as he advances rapidly 
over the first half of the text, and parallel with the work 
in the Common School Arithmetic is that carried forward 
in the Mental Arithmetic through Fractions. 

During this year all definitions should be learned and re- 
tained in the mind. Good definitions stand as " hitching- 
posts" to which knowledge must be tied. 

To understand the full import of this remark, the teacher 
should take simple addition, after the class has passed over 
the subject in the Common School Arithmetic, and arrange 
an outline as follows : 



r 1. Definition. 
2. Terms 



3 
o 






1. Parts. 

2. Sum, or Amount. 



' . 11. Plus. 

8 - Sl ^ 1S \ 2. Equality. 

r 1. Like numbers can be added. 

. . , 12. The sum is equal to all the parts. 

4. Principles. 1 g The gum . g the game in kind ag the partg 

1 4. Units of the same order are added directly, 
f 1. Writing the numbers. 

m ~ ,. ! 2. Drawing a line beneath, or making the sign of 

5. Operation. equalUy 

[ 3. Adding, reducing, etc. 

6. Exercises. 

7. Rule. 

8. Proof. 

Let a similar plan be followed after each subject is passed 
over, and the pupils will soon learn to arrange and classify 
quite accurately any subject that they study. This habit 
is invaluable, and it can be applied in other branches as 
advantageously as in Arithmetic* Such outlines, made by 
the pupils and corrected or modified by the teacher, afford 
excellent methods for reviews and analyses of topics. To 
pick up the scattered pieces of a subject and put them to- 
gether into a consistent whole involves all the constructive 
and reflective faculties of the mind. To this must be added 



52 ARITHMETIC. 

the requisite skill in asking pointed, searching, and appro- 
priate questions. The teacher who does not know how to 
ask questions at the proper time and in the right manner 
will never succeed in teaching any subject, much less Arith- 
metic. There should always be a plan in questioning 
either a pupil or a class, and that plan should usually 
be in an ascending scale. It is not enough that a pupil 
answers in a set form of words. The teacher must find 
out that the words used by the pupil are understood in 
their arrangement, and that they convey the correct idea 
to the learner's mind. What is said here lias reference to 
relevant questions and not to that thoughtless jabber and 
clatter sometimes heard in recitations. 

There are four things now that the pupil should learn : 
1. To hear correctly the conditions of a problem, or to 
understand a problem when he reads it. 2. To remember 
exactly all the conditions when a problem is heard or read. 
3. To think what is to be done. 4. To do it. 

Let these steps be firmly impressed upon the learner's 
mind, and he will then use his faculties to much better ad- 
vantage. As the knowledge in his mind is put into form, 
he is able to strengthen himself when he can pick out his 
own weaknesses. To grasp conditions with vigor, and to 
hold them tenaciously, are the first preliminary steps in tlie 
solution of a problem. These are followed by correct 
joning upon the data given. If a mistake is then made 
in the lino of thought afterwards, the inference is that the 
pupil started in the wrong direction. 

To find his mistake and to retrace his steps is one of the 
t important acts in the (earner's progress. The manner 
of doing the work is a matter of no small consequence. 
Everyone admires the beautiful clean page, free from all 
blemishes, whether it be a letter, a trial balance, a legal 
instrument, or what not. The same careful, exact work 
thai 3 elsewhere should he secured in mathematics. 



EXTENT OF THE FIFTH TEARS WORK. 53 

Since mathematics is an exact science, exact, neat, legible 
work is the only kind that should be accepted. 

Solutions lay under contributon language and taste in 
the arrangement and presentation of each thought. Work- 
ing questions for the sole purpose of "getting answers" is 
not worth much from the disciplinary side ; hence all 
written work should be neatly, legibly, and carefully done. 
The mathematical signs abridge ordinary language, and 
they are to be employed as often as possible in the solution 
of problems. 

Extent of the Year's Work. 

This year's work in the Practical or Common School 
Arithmetic for an average class will comprise a thorough 
review of the fundamental rules, fractions, denominate 
numbers, mensuration of some plane surfaces and of some 
solids. Stress ought now to be put on the " why" at each 
step. The teacher should be ready to ask " why" whenever 
the pupil appears to be in doubt; or better still, if the 
pupil will ask himself the question. To secure good work, 
let the teacher call for a written solution of some problem, 
and then keep the neatest and the best as specimens to 
stimulate others to greater effort. 

Teachers should avoid helping pupils and classes too 
much. Skillful questions by the teacher, not class-carry- 
ing, set the pupils to thinking, and are needed to arouse 
the latent energy of a class when all other efforts fail. 
Questions on each topic at first ought to be aimed low, — 
leveled to the learner's comprehension. Gradually they 
should take a higher and wider range. There is great 
danger in aiming too high at the beginning. The questions 
frequently go so far above the pupils' positive knowledge 
that vagueness and emptiness are succeeded by discourage- 
ment. 



ARITHMETIC. 



Definitions. 



Each subject lias its technical or arithmetical terms. 
These terms, if not already known, should be learned in 
connection with each topic as it is studied. The impor- 
tance attaching to cancellation, divisors, and multiples de- 
pends upon two things: 1. To shorten work; 2. To test 
results. Consequently they are to be mastered as helps. 

If a pupil is strong in fractions, he is well prepared to 
make proper advancement in his arithmetical studies; but 
till he masters fractions thoroughly, it is a waste of time 
for him to take up other topics. A good working knowl- 
edge is not sufficient. He must go to the root of the 
matter and master principles so thoroughly that when the 
teacher asks for the solution of a problem, it is given with- 
out hesitancy. Terms and processes must be understood, 
and thoroughly grounded in the mind. These objects are 
easily accomplished when one step is taken at a time. 

The subdivisions to be learned and remembered are: 
1. Definitions. 2. Classes of fractions as to kinds, as to 
value, as to form. 3. Terms, including numerator, de- 
nominator, similar, dissimilar. 4. Principles. 5. Keduc- 
tion of fractions. 6. Applications under addition, subtrac- 
tion, multiplication, division, etc. 7. The comparison of 
the treatment of fractions with the four fundamental rules. 

Simple exercises are to be used at first for illustrating 
new subjects. 

The connection between compound reduction and the 
reduction of fractions needs to be pointed out to the pupil; 
or in other words, he is better satisfied when he can trace 
relations himself. 

Eelated knowledge can be hitched together, and the 
more hitching the pupil can do, the better he is satisfied 
with Lis own work. With the "why" at each step, there 



MENTAL ARITHMETIC. 55 

should be another of " likeness" or " unlikeness" immedi- 
ately following. 

After common fractions, decimals should be studied with 
the same degree of thoroughness. Changing common 
fractions to decimals, and the reverse process, have a strong 
tendency in directing the pupil to observe the close con- 
nection existing between the two so-called species of frac- 
tions. Again, the reader is reminded that one problem 
worked out and well understood is worth pages of problems 
hastily sketched and partially comprehended. 

For every problem selected from the book, let the pupil 
make a corresponding problem of his own. Encourage 
each pupil to make his own illustrations and methods of 
solutions. Spend considerable time in having pupils read 
and interpret problems without requiring solutions. This 
will test their knowledge in interpretation. 

During this year's work the pupils should "clean up 
everything well as they go." % 

Mental Arithmetic. 

A general direction for solving problems in Mental 
Arithmetic will be given in this connection, and it may 
be employed advantageously in the part of the Third 
Grade and through the Fourth Grade, and it should be 
continued for obvious reasons as long as the pupil studies 
Arithmetic. 

Direction. — 1. The teacher will read or state the prob- 
lem once, slowly and distinctly. 2. The pupil, or class, 
will give the answer to the problem. 3. The pupil, or 
pupils separately, will give a short, connected, logical 
analysis, and a conclusion. 

To be avoided. — 1. Long, tedious analyses. 2. Letting 
the pupils use the text-book during recitations. 

The laws of numbers are abstractions derived from ab- 



56 ARITHMETIC. 

stractions. Comparing an abstraction with another abstrac- 
tion between which a relation can be affirmed or denied, 
brings out a conclusion clear and unquestionable to the 
pupil's mind. All such exercises develop the reasoning 
faculties if they appeal to the understanding. Mental 
Arithmetic, if properly taught, cultivates the reasoning fac- 
ulties more than any other of the common-school branches. 
It requires the pupil to hold each question firmly in his 
mind while he puts the conditions together and derives a 
necessitated conclusion. 

The Mental Arithmetic work during this period should 
be so arranged as to smooth the work in the Common 
School Arithmetic. In some cases it may very advanta- 
geously precede the written work. Each book should be 
studied separately and the lessons recited at different 
times, and if either must be slighted, let it be the Com- 
mon School Arithmetic. A bright pupil will learn all 
that is necessary to be known in Mental Arithmetic to Per- 
centage in four or five months ; but too frequently the 
"holding-back and long-drawn-out policy" is adopted 
under the mistaken notion that there is great danger of 
impairing the pupil's thinking faculties. Active, vigorous 
thinking never hurts or wears out a man, woman, or child, 
if proper physical exercise alternates with it ; but any 
amount of " rusting out " is induced by laziness. Let the 
teacher remember that Mental Arithmetic is to be pursued 
as a distinct and independent study. Never is it to be 
ciphered out, but it must be thought out. It is valuable 
to the pupil chiefly because it requires thought-work. 

It is not every book labeled " Mental or Intellectual 
Arithmetic" that deserves the name. There are weak 
books that require no effort on the part of the pupil. Of 
course little benefit can be derived from the study of such 
books. A year devoted to Mental Arithmetic when a boy 
or a girl is properly prepared to take up the subject and to 



SIXTH YEARS WOBK 57 

push it vigorously is worth more than twice the time de- 
voted to any other subject in the common-school course. 

Pupils naturally prefer Mental Arithmetic to Written. 
It affords a better field for the display of intellectual skill 
and superiority. Each solution carries a conscious convic- 
tion that is a tonic to the reasoning powers. 

In conducting recitations, quick, sharp, accurate work 
must be pursued. By gradual practice pupils will handle 
large numbers with almost as great ease as smaller numbers. 

Sixth Year's Work. 

Percentage and its applications, Ratio and Proportion, 
Evolution and Involution, the progressions, and mensura- 
tion, are the important subjects to be studied during this 
year. The same remarks in regard to mastering definitions, 
principles, and processes, made in the preceding year's work 
will apply to this grade. Percentage is simply a continua- 
tion of fractions. This idea is fundamental, and the sooner 
the pupil grasps it the more rapid headway he will make 
in the subject. To show the close connection between frac- 
tions and percentage is not a difficult matter. 
. For instance, -J- = .2 = 20 ■$. Here we have the same 
value expressed under three different forms, and in com- 
puting we may use any one we choose. 20$ of anything 
not only means "per cent/' but it means iVV? an abstract 
fractional number. The pupils should be so well drilled 
on the equivalent forms that one readily suggests the other. 
It is more convenient in practice to take -J- of a number 
than to find 16 1$ of it, and so of many other aliquot parts 
of 100. 

Practically the solution of problems in Percentage in- 
volves three cases : 1. To find the Percentage. 2. To find 
the Eate. 3. To find the Base. The pupil must drill long 
enough on each case to master it perfectly before beginning 
the next one. 



58 ARITHMETIC. 

Following the usual notation, the three cases are repre- 
sented thus: 

(1) B X R<f> = P. 

(2) P + Bi = B. 

(3) P + B = 72$. 

A problem under each case will now be given and solved. 

1. Find 40j< of 960 sheep. 

Solution. 100$ = 960 sheep ; 

1$ = 9.6 sheep ; 
40$ = 9.6 X 40 = 384 sheep. Ans. 

Second Solution. 100$*= f = 960 sheep ; 
20$ = \ = 192 sheep ; 
40$ = f = 192 X 2 = 384 sheep. Arts. 

This problem in common language means that f of 960 
sheep are to be found. It is a good exercise to have the 
pupils change problems from the percentage to the common 
form. 

The ciphering solution is given thus: 

960 X .4 = 384. Ans., which corresponds to formula (1). 

2. What per cent of $80 is $120? 

Solution. $120 is J- of $80; but f of any number is equal 
to 150$ of that number ; hence $120 is 150$ of $80. 

Or, -V/ = I = 15( ¥- An ^ b y formula (2). 

3. A paid B $20, which was 6£$ of what he owed B ; what 
was the debt? 

Solution. 6£$ = $20; 

1$ =$vs 

100$ =(i«^i^ = $320. Ans. 
Or, $20 -4- -jV = $320; or, $20 ^ .06£ = $320. Ans. 

If it becomes necessary to find the amount or difference, 



COMMISSION AND BROKERAGE. 59 

the formulas are B ± P = A or D. A when the upper 
sign is taken, and D when the lower sign is used. 

By extending the formulas to include the amount and 
difference, another case arises in practice. The following 
problem illustrates it : 

A sold a horse for $80 and gained 14f# on the cost price; 
required the cost. 



Solution. 


100$ + 14f$ = f = $80; 
4 = $10; 






% = $70. 


Ans. 


Or, 


$80 -f- 1.14$ = $70. 


Ans. 


Or, 


100$ + 14f£= 114f* = $80; 






100$ = -S;o. 


A ns. 



If the pupil keeps his mind clear upon the cost price of 
an article, he is not likely to encounter any serious difficulty 
in understanding how to classify and solve problems in 
Percentage. Encourage pupils to work a problem in two 
or three different ways after they know how to do it one 
way well. 

Commission and Brokerage. 

This subject is not always clearly presented in the text- 
books. There are two distinct classes of problems, and 
confusion often arises in the minds of the pupils because 
they are unable to classify the problems properly. The 
two classes of problems are : 

1. Those in which Commission is charged for investing; 
and 

2. Those in which Commission is charged for selling, or 
for collecting. 

The money handled by the agent in the business for the 
principal is the base. In the first class of problems the 
agent's commission should be deducted before he makes an 



60 ARITHMETIC. 

investment for the principal, and the agent should never 
charge commission on his commission. 

The second class differs from the first in this: the agent 
charges commission on all he sells or collects. 

From the foregoing it is evident that the agent's com- 
mission in the first class of problems is in the amount that 
is sent to him to be invested, and that he must deduct his 
commission before he invests the remainder for his prin- 
cipal. This gives rise to two processes, namely, the amount 
of commission the agent must receive, and the amount of 
money that the principal must transmit to include the 
agent's commission and the investment. Plainly, two trans- 
actions are involved : 

1. The amount sent to the agent, to find how much he can 
invest, less his commission. (1) 

2. The sum to be invested, to find how much must be sent 
to cover both commission and investment. (2) 

Under the second class there are two kinds of problems, 
as follows : 

1. The amount of sales, to find how much must be sent to 
the principal, less the agent's commission. (3) 

2. The sum sent to the agent, to find the amount of the 
sale. (4) 

Let the teacher select a problem under each of the pre- 
ceding conditions, and then tell the pupils to classify them 
as (1), (2), (3), (4). 

When they are familiarized sufficiently with the four 
kinds, let them begin with the solution of problems under 
each case. Commission should not be passed over until 
the pupils are able to discuss clearly and intelligently 
all ordinary problems in the text-books. If the distinc- 
tions are clearly marked and observed at the outset, Com- 
mission is easily understood; otherwise it is groping in the 
dark. 



INTEREST. 61 

Interest. 

As soon as the definitions are learned and the pupils have 
some knowledge of Interest, the teacher should borrow a 
statute from a Justice of the Peace, and let the class read 
the law on Interest and Usury. This brings forward to a 
certain extent the legal phase of the subject, and when 
promissory notes are spoken of, a good opportunity is 
afforded for teaching something of the law of contracts and 
the qualifications required of parties before they can legally 
make a contract. 

Interest from the mathematical standpoint may be ex- 
pressed under the most general form thus : 

P x B X T= Interest. 

Or, when expressed in words : 

The Principal multiplied by the Rate expressed decimally, 
multiplied by the Time in years, equals the Interest. 

While this is the general form, yet, in a large number of 
examples, the "6# method" possesses many advantages. 
It affords many excellent opportunities for analytical work, 
and lessens as well the labor in computation. Unless a 
problem is very simple, use the "6^' method." 

There are fewer mistakes when it is employed. To find 
the interest on $1 for any given time at G$ is always easy 
when the pupil once knows how, and then to find the 
interest on any number of dollars for the given time is a 
matter of simple multiplication. By all means perfect 
pupils in the "6$ method." For rapid, accurate work it 
is certainly superior to any other method of computing 
interest employed. However, the pupil should be perfectly 
familiar with the general method and then, according to 
the nature of the problem, he can use whichever is more 
convenient. 



62 AllITIIMETIC. 

Promissory Notes. 

Under this head the following terms should be explained 
and illustrated: 1. A Note; 2. A Negotiable Note, a Joint 
Nate: 3. The Maker, or Drawer; 4. The Holder, or Payee; 
5. Indorser; 6. Maturity; 7. Days of Grace; 8. Protest; 
9. A Partial Payment; 10. Indorsement. 

In actual practice one case in Interest is chiefly used, 

and that is to find the amount due on a note or account ; 

but in arithmetics all the cases arising from a consideration 

of the Principal, Hate, Time, Interest, and Amount are 

given. If any three of these five terms are given, the other 

two may be found. 

Discount. 

This is an obscure subject to the learner unless he gets 
the right start at the outset. The kind of discount that 
he studies most in his text-book is used very little in busi- 
ness transactions. To avoid confusion a correct definition 
of Discount should be given and illustrated. 

Discount is a deduction from a price, or from a debt. 

The kinds of Discount are three : 1. True Discount; 
2. Bank Discount; 3. Trade or Commercial Discount. 
True Discount should be sharply contrasted with Bank Dis- 
count. In True Discount, the debt does not draw interest 
except in rare cases. If the debt, however, bear interest, 
then the amount should be the sum discounted. Make this 
point plain to the pupil. The teacher should draw two notes, 
one not bearing interest and the other bearing interest, and 
let the pupil find the present worth of each ; then require 
the proof. Suitable questions will unfold the subject, 
and reveal its close connection with simple interest. 

Bank Discount. 

The clearest insight into Bank Discount is gained when 
pupils can go to a bank and see people depositing money 



BANK DISCO UNT—IN8 URANCE. 63 

or drafts, checking out, receiving certificates of deposit, 
sending money to distant points by means of drafts and 
checks, discounting notes, getting bills of exchange, etc. 

From almost any bank a teacher can get all the blanks 
necessary for illustrating the various transactions ordina- 
rily occurring there. The nearer the approach to actual 
business, the greater the interest the pupils will take, and 
the better the knowledge they will have of the subject. 

Here, again, contrast Bank Discount with True Discount. 
Bring out the agreements, and then the differences. Dis- 
count two notes of the same face value by each method. 
Let the notes bear interest, and then again take them as 
non-interest-bearing. Compare results. 

The class should not leave Discount until it thoroughly 
comprehends the subject. 

Trade Discount is so different from True or Bank Dis- 
count that confusion is not likely to arise. Yet it is well 
enough to call attention to it. 

Clearness in teaching depends entirely upon the grasp 
the teacher has on the subject. Clear, sharp-cut knowl- 
edge — how to communicate it, and how to interest pupils 
in the subject, will insure success. 

Insurance. 

This subject presents no serious obstacles, yet the dis- 
cussion in the Arithmetic can be greatly enlarged if the 
teacher will apply to "Insurance Agents" for blanks, cir- 
culars of information, and other printed matter. 

Then, there are the different kinds of Insurance, as well 
as the different plans of Insurance Companies, all of which 
should be studied. The statutes of each State and the re- 
ports of the Insurance Commissioners will throw a great 
deal of light on this branch of business. 

After a fire or cyclone, the teacher should explain how 



(U ARITHMETIC. 

the loss is adjusted, and, in case of a death, within what 
time the policy is paid. The true theory is to have the 
pupils study the subject as it is, not as it appears to be. 

Stocks. 

Begin the subject from the Daily Newspaper. Under 
the "Stock Exchange/' or " Money and Markets/' or 
"Bulls and Bears/' have the pupils read, the market reports 
and explain what they read. Such a report will most 
likely include Clearing House Statement, Total Transac- 
tions for the day or week, Bonds, Stocks, General and Local 
Markets. The newspaper is the mirror through which 
the pupils must read the volume of business as it is trans- 
acted at the great commercial centers. For every problem 
selected from the text-book, the pupils, or the teacher, 
should make two from the quotations given in the paper. 
The definitions used in Stock Exchange must be learned, 
or much of the language will be unintelligible ; also, the 
pupil needs to be reminded often of the difference between 
the par value of stocks and the market value, and on which 
of these brokerage, assessments, and dividends are computed. 
No new principle is involved in the solution of problems 
under this head. 

Taxes. 

This subject has a special interest because it reaches 
every home. Children, therefore, manifest more or less 
interest in those things which they hear discussed by their 
parents. For instance, it is a good thing for children to 
get some idea by whom and for what purposes taxes are 
assessed and collected, how they are used after collection, 
and many other questions that are incident to our system 
of revenue for national, State, county, municipal, town- 
ship, and school-district purposes. A good opportunity is 
afforded for teaching a wholesome moral lesson on giving 
in the valuation of personal and real property to the assess- 



COMPOUND INTEREST AND FOREIGN EXCHANGE. 65 

or. When pupils know what is meant, or intended to be 
meant, in a problem, it is not hard for them to begin it 
understandingly. 

Compound Interest and Foreign Exchange. 

Compound interest is, if the intervals are many, tedious 
to compute ; hence the Interest Tables found in many 
treatises. To these tables it is safer to refer when they are 
convenient. There is one difference only between simple 
interest and compound. Let the pupil mark it. Interest 
can become principal by agreement between the contracting 
parties. 

Foreign Exchange does not differ in principle essentially 
from Inland or Domestic Exchange. Tw T o or three points 
need emphasizing : 1. That a foreign bill is usually drawn in 
triplicate, called a set of exchange; 2. Simple Arbitration; 
3. Compound Arbitration. A copy of a Bill of Exchange 
should be shown to the class ; better still, secure a bill from 
a bank, and let the pupils examine it. Then require them 
to write bills. To illustrate Circular Exchange through 
different points, use cities in the United States as represent- 
ative commercial centers. Let the teacher explain the 
history of banking as it has been developed within the 
last few hundred years, and particularly compare the 
financial condition of our country with what it was just 
before the Revolution, just after the Revolution, and what 
it is at present. 

Ratio and Proportion. 

Definite knowledge is demanded in the treatment of 
Ratio and Proportion. Give the pupil a clear notion of 
what Ratio is. It must be plain to him; otherwise he is 
"shadow-hunting." New terms are introduced, and these 
he must learn too. The definition of Proportion follows 
from that of Ratio. Drill the class in Simple Proportion 



<*><; ARITHMETIC. 

till all the members can state problems correctly every 
time. The statement is the important point. Show how 
a proportion may be expressed in different ways, and that 
all are correct. Illustration: 4:8::6:12;or4-f-8 = 6-^12; 
or £ = {\ ; or 4 : 8 = 6 : 12, 

Explain the term Proportion. Submit many problems 
for the class to state in proper form. 

As a mental discipline, it is a strengthening exercise to 
have pupils analyze all the problems that they solve by 
simple proportion. This will show, too, that the same re- 
sult can be obtained in more ways than one. Aside from 
this, however, Proportion has a much wider application 
than this narrow view would seem to indicate, namely, its 
application in the other branches of mathematics. 

Compound Proportion is best taught by Cause and 
Effect. At least, my experience in teaching large classes 
is that pupils will learn it in much less time, and will 
seldom or never make mistakes in statements. In short, I 
regard the ordinary rule for stating questions in Compound 
Proportion as so much obsolete matter, retained in the 
books on account of its respectable antiquity. 

Practice a class upon easy exercises at first. Keep 
1st cause : 2d cause :: 1st effect : 2d effect 
on the board or slate to prevent mistakes and to test the 
work. 

Later on the work may be written at once for cancellation 
., 1C. 1 20. . . . ,.„ 1C. X 2E. , 

thus : 2E"k^ J and later still, 2C~X~TE~ = Answer ' 

Proportion is one of the easiest subjects in the Arithmetic 
when rightly presented. Many times it is made one of the 
most obscure. 

Square Root and Cube Root. 

There is no difficulty experienced in teaching pupils how 
to square, cube, or raise numbers to any required power. 



SQUARE ROOT AND CUBE ROOT. 67 

The process is that of simple multiplication; but to un- 
multiply the numbers is the troublesome part of the work. 
Evolution, then, is undoing what has already been done 
To show the reason for resolving a number into two, three, 
or any number of equal factors is now the duty of the 
teacher. This may be done in several ways, namely: 
1. The arithmetical method. 2. The geometric method. 
3. The algebraic method. For pupils studying arithmetic, 
a combination of the arithmetical and geometric methods 
is preferable for both square and cube root. To give direct- 
ness to the instruction in square root, after squaring a 
number, while resolving it into two equal factors, the 
geometric forms should be used to illustrate and to impress 
each step; in the absence of such forms, diagrams can be 
drawn on the blackboard. But both teacher and pupils 
can "whittle out" pieces of shingle that will answer the 
purpose very well, or cut out pieces of paste-board. Slices 
of a raw potato, apple, or turnip in case "of a pinch" 
make a fair substitute. However, all schools should be 
furnished with a full set of geometric forms. Some excel- 
lent teachers prefer to have the pupils take a square of 
either wood or paper and make additions to it before at- 
tempting to extract the square root of a number consisting 
of two periods. This paves the way for illustrating the 
numerical problem most advantageously. In combining 
the arithmetical computation with the geometrical con- 
struction, the learner's understanding is most effectively 
and permanently reached. If necessary let the pupil make 
three or four additions to the original square. Such work 
will delight him if the number whose root is to be ex- 
tracted is a large one. The area of the original square of 
each additional piece should be calculated separately. 

To facilitate work in square and cube root, the pupils 
should learn all perfect squares to four places of figures. 
This is not a difficult feat when a little reflection will show 



68 ARITHMETIC. 

in what figure each perfect power must terminate, and how 
easy it is "to square" mentally any integral number from 
1 to 99. Again, the simple method of forming a table of 
perfect squares by the addition of the successive odd num- 
bers reveals one of the most beautiful laws in connection 
with figures. More of this farther on. 

Cube Root should always be illustrated by using the 
blocks. First take a set of blocks for making one addition, 
preserving the form of a cube. Assume the edge of the 
cubical block as a definite number of inches, and then have 
its contents found. Next determine the length and thick- 
ness of each of the seven pieces that must be added to pre- 
serve the form, and then find the volume of each piece 
separately. This preliminary work must be continued till 
the pupils see why each step is taken, and can also explain 
the operation satisfactorily. When cube root is once 
rightly learned, it is never entirely forgotten. The pic- 
torial illustrations in the text-books, and such other devices 
as suggest themselves to the inventive teacher, will render 
the acquisition of this subject quite easy for the pupil of 
moderate arithmetical aptitude. 

I would not recommend the early introduction of the 
algebraic formula for the extraction of the cube root of 
numbers, although it may be introduced very appropriately 
farther on in the course. The safer method seems to be 
this : Each problem in cube root is to find the edge of a cube 
whose root must be the length sought ; or, given the volume 
of the cube, to find its edge. 

Series. 

As much as can be attempted under Series in this chapter 
is the presentation of a few of the very simplest cases. 
These should be applied chiefly to practical, or supposed 
practical, problems. The difference between Arithmetical 
Progression and Geometrical Progression ought to be 



MENSURATION. 69 

broadly outlined, and the distinction between a constant 
difference and a constant multiplier fixed in the mind and 
rationally apprehended by the pupil. A deeper insight 
may also be gained if some problems are proposed and 
solved which are related to compound interest and annui- 
ties. Anything like a satisfactory presentation of Series 
must be sought in our most advanced treatises on Algebra. 

Mensuration. 

There are so many ways in which measuring surfaces 
and volumes are applied, and such constant reference to 
books in order to solve common questions, that it is sur- 
prising how lightly the subject is touched upon ordinarily. 
Every boy ought to know how to find the contents of a 
piece of timber, having the dimensions given; or to tell 
how many bushels of corn a wagon-bed holds, and what 
deductions to make if the corn is not shelled; the number 
of cubic yards in a cellar or cistern; the number of tons 
in a hay-rick or a hay-mow ; the number of brick in a 
building or wall ; the number of gallons of water that a 
well, cistern, barrel, tub, or bucket will hold. Then there 
are formulas and rules for finding the area of circles, the 
surfaces and volumes of spheres, pyramids, cones, etc., 
that must be learned. Instead of solving such problems 
with the books open before them, pupils should know 
how to solve such problems, and commit certain data to 
memory ready for use. Some things are to be learned and 
retained because they are necessary and useful. To en- 
courage pupils to look beyond their arithmetics for the 
reasons involved in certain processes, the teacher can re- 
fer to the geometry in which demonstrations are found. 
The impetus given to the mind to look beyond the present 
boundary of knowledge is a powerful incentive, and when 
wisely directed leads to excellent results. 



70 ARITHMETIC. 

Miscellaneous Problems. 

All miscellaneous problems should be solved by the 
pupil. They are usually put in the book to test the pupil's 
knowledge. They are also intended as a review, and a 
"rounding up" of the subject. 

Outlines. 

Let the pupil now make a complete outline of the sub- 
ject as it is presented in his book. This will give him a 
comprehensive view of arithmetic, and it puts his knowl- 
edge into good form for wielding it with the least expendi- 
ture of vital energy. 

Mental Arithmetic. 

During this year the Mental Arithmetic should be thor- 
oughly completed. Whether a class commenced the year's 
work with percentage or the miscellaneous problems pre- 
ceding percentage, every difficult problem should be solved 
before beginning a new collection. Commencing the mis- 
cellaneous collection first, then the following kinds of 
problems will help to strengthen the reasoning faculties: 

1. Given the sum or the difference of the parts of a 
number, increased or decreased by a certain number, to find 
the original number. 

2. Given one part of a number of times another, or one 
part of a given number more than another, or the number 
of times one part equals a number of times another, or the 
part proportional to a given number, to find the required 
number. 

3. Problems in Proportion. 

The right kind of work in Mental Arithmetic now not 
only gives a mastery of the Arithmetic itself, but it also 
lays the foundation for future excellence in Algebra. 



MENTAL ARITHMETIC. 71 

Under Percentage and Interest the same care in the 
solution of problems should be observed as was indicated 
in the work of the preceding year. Mental Arithmetic 
and the problems solved are not to be Written Arithmetic. 
Work without pen or pencil is demanded. No matter 
how complicated a problem appears to be, it must be in- 
vestigated strictly upon the mental basis. The best results 
are thus obtained. No deviation from the plan outlined 
ought to be tolerated. Let all the cases in Percentage and 
Interest be mastered, and the pupil's progress in these 
rules ever after will be easy and rapid, however difficult the 
problems may appear. The analytic habits of mind thus 
inculcated will prove invaluable in other directions. 

There are other problems, however, that demand consid- 
erable attention, such as the pasture problems, beggar and 
number problems, labor problems, fish problems, pursuit 
problems, stock problems, horse and saddle problems, gam- 
ing problems, time problems, age problems, step problems, 
partnership problems, involution and evolution problems, 
will problems, and miscellaneous problems. All these prob- 
lems except the miscellaneous ones should be studied in 
classes; that is, all fish problems should be studied and 
solved consecutively. When commenced, the particular 
class should be completed before any other problems are 
taken up. In Mental Arithmetic all these special problems 
ought to be placed in groups or classes. Two American 
authors have adopted this idea in the classification of these 
problems— Dr. Brooks and Mr. George E. Seymour. 

Problems have two phases; one the arithmetical, and the 
other the graphical. These two make the deepest and 
clearest impressions when combined. When the relations 
are thought out and expressed, the next step is for the 
pupil to draw a representation of the conditions as ex- 
plicitly stated, and then to derive the implied conditions. 
As an illustration, suppose a fish problem is given for solu- 



7'2 ARITHMETIC. 

tion. After the problem is solved, then let each pupil 
draw a picture of a fish representing all the given condi- 
tions, and show from these how the required conditions are 
found. The illustrative process should not precede the 
solution. It is not necessary to illustrate every problem, 
but the process of illustrating should be so well understood 
that the pupils can devise illustrations whenever needed. 
Some pupils require graphic representations in order to 
seize all the conditions of a difficult problem, and in such 
cases the picture is an aid to the pupil. 

One problem of a class well understood by the pupil pre- 
pares him admirably for solving all similar questions. But 
it is oftentimes necessary to vary from the typical problem 
to elucidate a certain feature which is not quite clear to the 
pupil's mind. Of these variations the teacher is the judge. 
As a general thing, if the pupil can solve the most difficult 
problems, he will take care of the easy ones. If the habit 
is once formed by the pupil of going back and coming up 
again when he finds a " tough customer" until he succeeds 
in solving it, his success in Mental Arithmetic is assured. 



Advanced Arithmetic. 

Under this head will be given problems, principles, and 
comments. 

Rapid Methods of Adding. 

1. To add from left to right. 

74 

I. yj Explanation : GO + 70 + 13 = 143. The eye is 

accustomed to pass from the left to the right; hence we 
iiv (JO + 70 = 130, and 130 + 13 = 143. 



RAPID METHODS OF ADDING. 73 

Or we can increase 69 to 70, then add, and subtract 1 
from the sum. 

This process of adding two or more numbers should 
always be performed mentally. 

Let it be required to add a column consisting of 85 + 
97 + 68 + 73 + 56. Adding the tens first, we have 350; 
next the units, 29; hence, 379. 

2. To add columns in groups of 10, 20, 30, etc. 

This process consists in placing a small figure on the 
margin of the column, indicating the number of units in 
excess in each group. Accountants use this method, be- 
cause if stopped when adding a column the small digits 
along the margin indicate the work as far as the last group 
added. The same work is performed by those mentally 
who touch the thumb and forefinger when the first group is 
reached, the second finger at the second group, and the 
open hand at the fifth group, and so on, repeating. Groups 
of " 20 " appear most convenient in practice. " Practice 
makes perfect " is the motto. 

3. To add two or more columns at the same time. 

Here, again, it is most convenient to add from left to 
right. The addition is performed mentally. 

Thus: 96 

183 87 

45 228 
297 69 

_78 

375 
Explanation. By a previous method 183 is found; 45 
added gives 228; 69, 297; 78, 375. Or it may be done 
thus: 

90 + 80 = 170; 170 + 13 = 183; 183 + 40 = 223; 
223+ 5 = 228; 228 + 60 = 288; 288+ 9 = 297; 
297 + 70 = 367; 367 + 8 = 375. 
The entire mental work is expressed in detail. 



74 ARITHMETIC. 

4. To add three columns proceed thus : 

463 
921 

13 84 

Conceive the hundreds column to be separated from the 
other two columns; then add and combine. 

5. To add four columns at once. 

Conceive the two left-hand columns separated from the 
two right-hand columns; add the left-hand ones separately, 
then the right-hand, and combine the results. Thus : 

4862 
5978 
8321 



190 161 
Combining we have 1916L 

Some Contractions in Multiplication. 

1. To multiply by 5. Annex a to the multiplicand, and 
divide by 2. 

2. To multiply by 15. Annex two O's to the multiplicand, 
and to the result add half itself. 

3. To multiply by 25. Annex two 0% and divide by 4. 

4. To multiply by 75. Annex two 0% and take i of the 
result; the remainder is the product. 

5. To multiply by 125. Annex three O's, and divide 
by 8. 

6. To multiply by 175. Annex two 0% multiply by 1, 
and divide by 4. 

7. To multiply by 275. Annex two O's, multiply by 11, 
and divide by 4. 

8. To multiply by 166|. Annex three O's, and divide 
by 6. 

9. To multiply by 250. Annex three O's, and divide by 4. 



SOME CONTRACTIONS IN MULTIPLICATION 75 

10. To multiply by 333^. Annex three O's, and divide 
by 3. 

11. To multiply by 375. Annex three O's, multiply by 
3, and divide by 8. 

12. To multiply by 625. Annex three O's, multiply by 
5, and divide by 8. 

13. To multiply by 666f. Annex three O's, multiply by 
2, and divide by 3. 

14. To multiply by 750. Annex three O's, multiply by 3, 
and divide by 4. 

15. To multiply by 875. Annex three O's, multiply by 
7, and divide by 8. 

16. To multiply two numbers when the sum of their units 
equals 10 and their tens figures are alike. 

Thus: 84 X 86. Multiply the units and write the re- 
sult; multiply 8 by 9, and combine. 

Operation. 6 X 4 = 24; 8 X 9 = 72; hence, 7224. Ans. 

17. To multiply a number less than 100 by itself. 

Thus: 75 X 75. Multiply the units, and write the re- 
sult; then multiply 8x7, and combine. 

Operation. 5 X 5 = 25; 8 X 7 = 56; hence, 5625. Ans. 

When the number does not terminate in 5, proceed thus: 
68 X 68. The difference between 68 and 70 is 2; then 
2x2 = 4, and 6x7 = 42. Since the tens place is vacant 
in the first product, we write 4204. Ans. 

18. To multiply when the tens figures differ by 1 and the 
sum of the units equals 10. 

Thus: 86 X 94. 

Operation. 90 X 90 = 8100, and 4 X 4 = 16; hence, 
8100 - 16 = 8084. Ans. 

Always square the unit figure of the greater number and 
subtract. 

19. To multiply in a single line two numbers such as 
232 X 238. 



70 ARITHMETIC. 

Operation. 2 X 8 = 16; 23 X 24 = 552; whence, 55216. 
Ans. 

20. To multiply any number of two figures by 11. Write 
the sum of the two figures between them. Thus: 54 X 
11 = 594. 

21. To square any number of 9's instantaneously. Write 
as many 9's less one as there are 9's in the given number, 
an 8, as many ? s as 9 ? s in the product, and a 1. Thus : 
(9999999) 2 = 99999980000001. Ans. 

22. To square fractions of the form 6J, 1\, 8|, etc. Mul- 
tiply the whole number by the next higher digit, and an- 
nex £. Thus: (6i) 2 = 7x6 + i = 42£. 

23. To multiply two like numbers when the sum of their 
fractions equals 1. Thus: 8{ X 8f = 8 X 9 + | x f = 
72f Again, 39f X 39^ = 39 X 40 + \ X f = 1560f 

24. To multiply any number, as 5£, by itself. Thus: 5^ X 

O -f- 4 A 4 — ~'T6" # 

25. To multiply two numbers mentally. Thus: 327 X 58. 
Here 327 = 300 + 20 + 7, and 58 = 50 + 8. 

Begin at the left. 300 X 50 = 15000 

20 X 50 = 1000 

7X50= 350 

300 X 8 = 2400 

20 X 8 = 1G0 

7X8= 56 

18966 

Separating numbers and multiplying as in the last ex- 
ample is a valuable exercise. Any one who will practice it 
a short time will be astonished at the ease with which such 
operations are performed mentally. The advantage con- 
sists in working with numbers which the mind easily re- 
members. 

The following are some difficult arithmetic problemswhose 
solutions will be very acceptable to a large class of readers. 



MISCELLANEOUS PROBLEMS. 77 

1. A brewery is worth 4$ less than a tannery, and the 
tannery 16$ more than a boat; the owner of the boat has 
traded it for 75$ of the brewery, losing thus $103; what is 
the tannery worth? 

Solution. Let 100$ = the value of the tannery. 
Then 96$ = " " " " brewery, 

and 100 -f- 116 = 8Gj%fo = the value of the boat. 

By the problem, 

75$ x 96$ = 72$ = 75$ of the brewery; 
.-. 86^%fo - 72$ = 14^$ loss; 
1$ = $7.25; 
100$ = 8725. Am. 

Another Solution. §4 of the tannery is worth the brew- 
ery, and the boat is worth |J| = §§ of the tannery. 

But the brewery is worth | of ff = || of the tannery. 
Now if |f of the tannery is traded for only |f of the tan- 
nery, the loss equals f| — 1| = iff* This loss is §103; 
hence |f| = 8103, and ^ ¥ = $1; .-. $f§ = $1 X 725 = 
$725. Ans. 

2. My agent sold my flour at 4^ commission; increasing 
the proceeds by $4.20, I ordered the purchase of wher 

2$ commission; after which, wheat declining 3|$, my whole 
loss was 85; what was the flour worth? 

Solution. Commission on 84.20 = .08^-, and 84.11f| = 
amount invested in wheat, $4.11ff X 3$ = 80.13|£ de- 
cline on wheat. Total loss on 84. 20 = $.214$; 85-8.21 £f = 
84.78/ f = loss on flour. 

Let 100^c = value of flour, and 96$ = proceeds. Then 
96$ X y§ 2 = H4^ = commission for buying wheat; 96$ 
— 1|5.$ = 94^-$ = investment in wheat; and 94 T 2 T $ X 
3|.$ = 3^ T $ = loss on wheat. Total expense and loss on 



78 ARITHMETIC. 

wheat = 4$ + IWo + 3^ T $ = 9^ = $4.78^ T ; 1$ = $.53; 
100$ = $53. Ans. 

3. W. T. Baird, through his broker, invested a certain 
sum of money in Philadelphia 6's at 115|$, and three 
times as much in Union Pacific 7's at 89-|$, brokerage \<f> 
in both cases; how much was invested in each kind of 
stock if his annual income is $9920 ? 

Solution. Let 100$ = face of the Philadelphia stock, 
and 116$ = cost of the same; then 348$ = cost value of 
Union Pacific stock; 348$ -f- 90 = 3.86|$ = face value of 
the same. 6$ = income of the former, and 27 I 1 -^$ = 
income on the latter; but 6$ + 27 T ^$ = 33 T ^$ = $9920; 
1$ = $300, and 116$ = $34,800; whence 348$ = $104,400, 
cost of the latter. 

4. The Mutual Fire Insurance Company insured a build- 
ing and its stock for two thirds of its value, charging If $. 
The Union Insurance Company relieved them of one fourth 
of the risk, at L|$. The building and stock being destroyed 
by fire, the Union lost $49,000 less than the Mutual; what 
amount of money did the owners of the building and stock 
lose? 

Solution. Let 100$ = value of property; 66|$ = risk; 
lf$ X 66f$ = 1£$, premium on value of the property; 
\ of 66|$ == 16f$ = Union risk; 66f$ - 16f$ = 50$ = 
MutuaFs risk; 1-|$ X 16f$ = \i<> = Union premium; 1|$ 
_ i$ — J,^$ = MutuaFs premium; 50$ — \\$> = MutuaFs 
loss; 16f $ - \i — 16f%$ = Union's loss; 49 T V$ — 16^ 
= 32|$ = $49,000, excess of MutuaFs loss; 1$ = $1500, 
and 100$ = $150,000. 

100$ — 66|$ + 11$ = 34|$ = owner's loss. 
.-. 34^$ = $1500 X 34£ = $51,750. Ans. 



MISCELLANEOUS PROBLEMS. 79 

5. A merchant gives his note, 10$ from date, for 
$2442.04; what sum paid annually will have discharged 
the whole at the end of 5 years ? 

Solution. The interest due on $2442.04 at the end of 
the first year is $244,204; the principal must be diminished 
regularly by payments which are 10$ more on each dollar 
every year. 

First year, $1.00 

Second year, $1.10 

Third year, .'.91.21 

Fourth year, $1,331 

Fifth year, $1.4641 

Total paid, .... $6.1051 
To find the amount of the first payment, divide $2442.04 
by $6.1051 = $400. But $400 + $244,204 = $644,204, 
which =: the amount to be paid each year. 

6. Hiero's crown, sp. gr. 14|, was of gold, sp. gr. 19}, 
and silver, sp. gr. 10-J-; it weighed 17£ lbs.: how much 
gold was in it ? 

Solution. The crown displaced T -f- T of its weight in 
water; gold T 4 T , and silver ^ T . Hence we must compare 
the combining weights with the weight of the crown in 
water. 

_8 (21) ( 27027 ) or 74 
117 i _£ I i 444 I 121 

( 77 ) ( 27027 ) 195 

The combining ratios are T \ 4 3 and }f| ; 

17.5 X 74 ___ .- 

or — = 6ff silver, 

- 17.5X121 1Afi , ,. 
and — = lOff gold. 



80 



ARITHMETIC. 



7. A dealer in stock can buy 100 animals for $400, at 
the following rates: calves, $9; hogs, $2; lambs, $1. How 
many may he take of each kind? 

Solution. It is evident that the mean average price is 
84; also, that to sell a calf for $4 that is worth $9 is a loss of 
$5; while $2 is gained on each hog, and $3 on each sheep. 
Since the mean average price is $4, it follows that the 
losses and gains must balance, and such numbers must be 
taken as will balance the differences in prices. 





Operation. 




1 


3 


5 




10 


2 


2 




5 


60 


9 


5 


3 


2 


30 






8 


7 


100 



Explanation. The first column represents the prices, 
the second column the gains and losses, the third column 
balances the losses and gains when the highest and lowest 
priced animals are combined, and the fourth column when 
the calves and hogs are combined. The sum of the third 
and the fourth columns is 15; but 15 will not divide 100, 
hence we must find 8 times some number plus 7 times 
some number that will equal 100, or 8 times a number 
plus 7 times some other number = 100; whence 8x2 + 
7 X 12 = 100; therefore, multiplying the separate num- 
bers in third column by 2, and the separate numbers in 
fourth column by 12, we get 10, 60, and 30 for one set of 
answers. If we take 9 instead of 2, and 4 instead of 
12, we have 45, 20, 35. 



MISCELLANEOUS PROBLEMS. 81 

8. A hall standing east and west is 46 feet by 22 feet, 
and 12| feet high; what is the length of the shortest path 
a fly can travel, by walls and floor, from a southeast lower 
corner to a northwest upper corner? 

Solution. Turn the west wall down flat with the floor 
outward; this will form a rectangle 58^ feet in length and 
22 feet in width, whose diagonal = V (58J) 2 + (22) 2 = 
62 -f- feet. Turning it down to the north side of the floor, 
the sides of the rectangle are 22 + 12^ = 34| feet and 46 
feet, whose diagonal = V (34|-) 2 +"(467 = 57|- feet. Ans. 

9. How many stakes can be driven down upon a space 15 
feet square, allowing no two to be nearer each other than 
1| feet; and how many allowing no two to be nearer than 
li feet? 

Solution. 1. It is evident that eleven rows of stakes can 
be set "in squares," having 11 stakes in each row; but it 
remains to be shown that this is not the greatest number 
of stakes that can be set on the plat of ground. If we put 
11 stakes on the bottom margin or first row, and if w T e 
place directly above the middle point of the distance 
between any two consecutive stakes a point at the required 
distance from the two points, forming an equilateral 
triangle, the second row will contain just 10 stakes, and 
the distance between the first row and the second row = 



V(iY - (I) 2 = 1.29903 + feet. Hence the gain in width 
is 1.5 - 1.299903 + = .20097 feet, and to gain an addi- 
tional row seven rows must be set. Of the first nine rows, 
the first, third, fifth, seventh, and ninth will have 11 stakes 
each, and the second, fourth, sixth, and eighth, 10 stakes 
each. In the three remaining rows nothing can be gained, 
so these will have 11 stakes each. Therefore, 11 X 8 — 88, 
and 10 X 4 = 40, and the greatest number is 128. Ans. 

2. Arranging in squares, 169 can be set. Proceeding in 
the first part of the problem, we have five rows of 13, four 



82 ARITHMETIC. 

rows of 12, and five rows of 13 stakes, making a total of 
178 stakes. If this is the greatest number, then the 
problem is solved with fourteen rows. Let us see if 
fifteen rows can be set. Now make the rows from the base 
line alternately reach the left side, and the second set of 
lines alternately reach the right side, and fifteen rows of 12 
stakes in each row can be set = 15 X 12 = 180 stakes. 
Remark* This last method is regular "lattice- work." 

10. A wooden wheel of uniform thickness, 4 feet in 
diameter, stands in mud 1 foot deep; what fraction of 
the wheel is out of the mud ? 

Solution. In the annexed diagram let AC = BO = DO 
= 2, the radius of the wheel; OE = DE = 1; then AE = 
\T% =: 1.732 +; AB = 2V1$ = 3.464 + feet. By the 
rule given, for finding the area of 
the segment of a circle, the area of 
the segment in this case = (l 3 -f- 
6.928) + f(3.464 X 1) = .1443 + 
2.3094 = 2.4537. But the area of 
the wheel = 12.5664, and the area 
above the mud = 12.5664 - 2.4537 = 
10.1127; whence 10.1127 -5- 12.5664 = 
.804 + of the wheel above the mud. 

11. By discounting a note at 20 fo per annum, I get 
22£# per annum interest; how long does the note run? 

Solution. Discount = 20$ of the face = 22J$ of the 

20 
proceeds; proceeds = — = f of the face; discount = \ 
&&\ 

of the face = 11£$; time = — * = | year = 200 days. 

12. A 12-inch ball is in the corner where walls and floor 
are at right angles; what must be the diameter of another 




MISCELLANEOUS PB0BLEM8. 83 

ball which can touch that ball while both touch the same 
floor and the same walls ? 

Solution. . There are evidently two problems in this one: 
(1) When the ball is in the corner behind the 12-inch 
ball; (2) When it is in front of the 12-inch ball. 

1. The distance from the corner of the room to the 
center of the 12-inch ball is the diagonal of a cube whose 
edge is 6 inches, i.e. the diagonal = |/(6 2 + 6 2 + 6 2 ) 
= 6 V3; the distance from the nearest point of the ball to 
the corner = 6 V 3 — 3, and from the farthest point = 6 

V~3 + 6. Now assume the radius of the required ball to 
be 1; then the point at which it touches the given ball is 

V3 + 1. By the problem this distance is 6 1/3 — 6; 
hence (6 V~l - 6) -4- ( V~3 + 1) = 6 ( 2- VH) = 1.6077 + 
= radius of required ball, and its diameter = 3.2154 inches. 

2. In this case we have (6 Vd + 6) -^ ( VH — 1) = 6 
(2 + Vl) = 22.3923 inches, and the diameter == 44.7846 
inches. 

13. A workman had a squared log twice as long as wide 
or deep; he made out of it a water-trough, of sides, ends, 
and bottom each 3 inches thick, and having 11772 solid 
inches; what is the capacity of it in gallons ? 

Solution. Cut the log into two equal cubes The hollow 
part of each of these cubes = 5886 cubic inches. A 
cubical box having sides 3 inches thick may be regarded as 
made up of 6 square blocks, 12 blocks of the same length, 
and 3 inches wide, and 8 corner cubes of 27 inches each. 
Take away two of the square blocks and one long block, 
and the half-trough is left, containing 4 square blocks, 11 
long blocks, and 8 cubical blocks, making a total volume of 
5670 cubic inches. Now if we take a side surface in each, 
there are 4 squares and 11 strips = 1890 inches in area. 



84 



ARITHMETIC. 



One square + V - strip 3 inches wide = 472^ square 
inches. Divide the strip *£- of 3 inches wide lengthwise 
into '2 strips e::ch 4^ incli3s wide and placed on adjacent 
sides of the square, and there will still be lacking a square 
(4J) 1 = 17^j square inches. Therefore the completed 
square must contain 472^ square inches + l^Vr square 
inches = 489ff square inches; but ^48911 = 22J- inches. 
Consequently, 22-J- — 4-J- = 18 inches, which is 6 inches 
less than the thickness of the log. Hence the log is 48 
inches long and 24 inches wide. The volume of the 
trough = (2 X 13 X 18 X 21) -f- 231 = 68 T 8 T gallons. 



VN 



14. A tin vessel having a circular mouth 9 inches in 
diameter, a bottom 4^ inches in diameter, and a depth 
of 10 inches is J part full of water; what is the diameter 
of a ball which can be put in and just be covered by 
water ? 

Solution. Let the annexed figure represent a middle 
section of the vessel, being the 
center of the ball which is just 
covered with water, and P being 
the center of a ball that will just 
drop into the vessel and is level 
with the top of the vessel; the 
line DE represents the lower part 
of the vessel. Conceive the vessel 
to be extended to C, forming a 
regular cone. By the problem, 
AN = BN = 4£ inches ; NM = 

10 inches = MC. Then AG = ' 

|/20 2 + (4£) 2 = 20.5 inches. The area of the triangle ABC 
— 90 square inches, and its perimeter = 50 inches. It is 
shown in geometry that the radius of an inscribed circle == 
the area of the circumscribing triangle divided by half the 




MISCELLAtfEOVS PROBLEMS. 85 

perimeter; therefore the radius NP = 90 ~ 25 = 3.6 
inches. 

If GRH represents the surface of the water after the 
sphere, 0, is dropped into it, there are given the cone NC 
and its inscribed sphere, P, and its radius, NP, to find the 
radius FO of the sphere inscribed in the cone RC. This 
involves the principle of similar solids. 

Volume of cone ABC = (4i) 2 X *£ X n = 135tt; 
Volume of sphere NP = (7^) 3 X i?r = 62. 208 it. 

The ratio of the sphere to the cone = -§-§-§-, if we consider 
the cone as 1, then the volume of the cone DEC = 
(i) 3 = 4, since MC is half jVCI The volume of ABDE = 
1, and 7T = 3.1415926. 

The volume of water is \ of \ = %\, but the volume of 
the cone ABC— the volume of the sphere NP = 1 — ■ 
HI = HI, and the volume of the cone GHC — the volume 
of the sphere FO = £ of cone DEC -\-^ of the water = £}. 

Similar solids are to each other as the cubes of their like 
dimensions; hence 

111 -U''- (3.6) 3 : (3.09836) 3 = {F0)\ 
whence 3.09836 x 2 = 6.1967 inches, required radius. 

15. How many inch balls can be put in a box which 
measures, inside, 10 inches square and is 5 inches deep ? 

History. This problem is due to Dr. U. Jesse Knisely, 
whose solution created no little controversy owing to his 
method, which was different from that given by all preced- 
ing arithmeticians. With his solution discussion began. 
As the problem was more carefully examined, other solu- 
tions were published. 

The appended solutions indicate the progress- of the 
investigation. 



86 ARITHMETIC. 

This is one of the most remarkable arithmetical problems 
eyer published. 

Dr. Knisely's Solution is as follows : Place on the bottom 
100 balls; above these, 81 balls; then 100, then 81, and so 
on as high as possible without entering the fifth inch of the 
height. The top of the second and each subsequent layer 
so placed is .7071 inch higher than the preceding. With- 
out passing into the fifth inch there can be, thus placed, 
100 + 81 + 100 + 81 + 100 = 462 balls, and occupying 
:].^IS inches of the depth; there remains a space 10 inches 
square and more than 1 inch deep. How many balls can 
be placed in this space ? Starting with a row of 10, and 
arranging triangularly, so as to have rows of 10 and 9 alter- 
nately, the second and each succeeding row will extend 
^V3, or .866 inch farther than the preceding; hence 
there could be eleven rows, six 10 ? s and five 9 ? s, or 105. 

But if we let the triangular arrangement cease with the 
ninth row, there will be five rows of 10 and four rows of 9, 
making 86 balls, occupying 1 inch. .866 inch X 8 = 7.92 
inches of the width; there remains a width of 2.07 inches, 
which will allow two rows of 10. Then, 86 + 20 = 106, 
and 462 + 106 = 568. Ans. 

Second Solution, by Mr F. W. Brown, Byer, Ohio. He 
places four rows of 9 balls each on the 5 X 10 inches base 
of box, having the outside rows against the sides and a 
space of 1 -r- 3 = \ of an inch between two contiguous balls 
across the box and 1 -f- 8 = i of an inch lengthwise. This 
makes 36 balls in bottom layer. On this he places three 
rows of 8 balls each, so that one touches 4 balls in bottom 
layer. By a calculation he makes the balls of the second 
layer extend just below the plane passing through the 
center of the balls of the first layer; hence the second 
layer extends upward not quite half an inch above the 
top of the first layer. The third layer is placed directly 



PROBLEMS; 87 

on top of the first, and the fourth on the second, and so 
on till nineteen layers are thus put into the box. 

The first layer is repeated nine times and the second eight 

% ( (4 X 9) X 10 = 360 ) 
times, giving a total of •< and [ 576 balls. 

( (3 X 8) X 9 = 216 ) 

Third Solution, by Mr. F. F. Vale of Ohio. He puts in 
100 on the 10 X 10 base; then 81 in second layer, 100 in 
third, 81 in fourth, and 100 in fifth, making 462 balls. The 
sixth and seventh layers are placed in such a way that five 
rows of 9 balls and one row of 10 balls are in the sixth, or 
55 balls, and six rows of 10 balls, or 60 balls, in the seventh. 
He makes the centers in fifth layer 4 X .797107 = 2.828428 
inches above centers in first layer, and 4 — 2.828428 = 
1.17157 below centers of the seventh layer. 

Hence the total number is 462 + 115 = 577. Ans. 

The 'Fourth Solution was obtained by Mr. A. J. Trapp, 
Pleasant Hill, Missouri. If the box were but 1 inch deep, 
it manifestly would hold only 50 balls. These would 
just fill the box if arranged in ten rows of 5 balls each, 
the centers of adjacent rows being 1 inch apart. But 



, row. 
.. row, 


o 


2 


/ JB \ & / 


CO 


\^ 




b 




xSv/'X^'' i 


i 




jr^^\ # j 


•^cT^N, 




v y^ 







ab — \ inch; 
«[ 5x b>-<^ A £>■< c5=iinch; 



Vl* -£* = . 86603 



Side of box. 
Fig. l.PJan of 1st layer. 



88 ABITHMETIO. 

if the balls in the even rows (second, fourth, sixth, etc.) 
are placed in the cavities between balls in the odd rows 
(first, third, fifth, etc.), the centers of adjacent rows will be 
only .86603 — inch apart, and eleven rows can be placed in 
the box, the odd rows having 5, and the even rows 4 balls, 
and an unoccupied space of .339 -f inch will be left between 
the outside of the eleventh row and the end of the box; that 
is, the 50 balls so arranged occupy a space 1 inch high, 5 
inches wide, and 9.661 inches long, their aggregate volume 
filling t 5 o 4 -q of this space. This is evidently the most compact 
arrangement possible for a single layer of balls. 

Now as to the most compact arrangement for two or more 
layers. If a ball were placed on top of each ball in the first 
layer, the two layers would contain 100 balls, and would 
occupy a space 2 X 5 X 9.661 inches; the aggregate volume 
of balls filling -£fo of the total volume of space as before. 
If, however, the balls in the second layer are placed in the 
trihedral pits between the adjacent rows in the first layer, 
the plane passing through the centers of balls in the second 
layer will be but .8165 inch above a similar plane in the first 
layer, and the two layers will go into a box 1.8165 inches 
high. 

cd = 1 inch; 
Wlayer. ac = ad — .86603 inch. 
^n. area acd = .353556 +• 

Snd.layer. and 

• 353556 +^=,8165- 



ac 

also, 



^>/N XV y Vad*^db 2 =ab=.289-. 

Bottom ot box. 
Fig. 2. 

Since the centers of the rows in the second layer are .289 
inch to the right of the center of the corresponding row 



MISCELLANEOVS PROBLEMS. 89 

in the first layer, and since the outside of the eleventh row 
in the first layer is .339 inch from the end of the box, 
there is .339 —.289 = .05 inch more than enough room 
in the box for eleven rows in the second layer; that is, the 
two layers occupy a space 1.8165 inch high, 5 inches wide, 
and 10 — .05 = 9.95 inches long. But the second layer 
contains only 49 balls — the six odd rows containing 4 and 
the five even rows 5 balls each; therefore the aggregate 
volume of the 99 balls in the two layers fills more than 
T %\ of the volume of the space occupied. Hence this is the 
most compact possible arrangement of layers, the loss of 
one ball in the even liyers being more than compensated 
by the gain in space caused by decreasing the height be- 
tween all the layers to .8165 inch. 

Since the depth of the box is 10 inches, it will contain 

twelve layers ( n 1fr + 1]* leaving .0185 inch to spare 

above the upper layer. 

The six odd layers contain 50 balls each = 300 
The six even " " 49 " " = 294 

Total number of balls in box, 594 

The aggregate volume of the 594 balls fills ^& of the 
total volume of the box. The same number of balls could 
be placed in a box 5 X 9.95 X 9.9815 inches, and would 
occupy T 6 ^ 



16. My tailor informs me it will take 10^ square yards 
of cloth to make me a full suit of clothes. The cloth that 
I am about to purchase is If yards wide, and on sponging 
it will shrink -£$ in width and length. How many yards 
of the above cloth must I purchase for my "new suit"? 



90 ARITHMETIC. 

Solution. The author designed this problem to be 
solved by proportion. Whence 

20 : 19 : : 1.875 yds. : 1.78125 yds. = width after sponging. 

Again, 

20 : 19 : : 1.78125 yds. : 1.6921875 yds. = second shrinkage. 

Consequently, 1.6921875 yds. : 1 :: 10.25 : 6 T ff T yards. 

Ans. 

17. Suppose a clock to have an hour-hand, a minute- 
hand, and a second-hand, all turning on the same center. 
At 12 o'clock all the hands are together and point at 12. 

(1) How long will it be before the second-hand will be 
between the other two hands and at equal distances from 
each ? 

(2) Also before the minute-hand will be equally distant 
between the other two hands ? 

(3) Also before the hour-hand will be equally distant 
between the other two hands? 

Solution. Let the annexed diagrams represent the posi- 

T12£ 

m 




FUj. 1 Fiy, 3 FUj. £ 



tions of the three hands respectively when they first satisfy 
the conditions of the question. When the hour-hand 
moves over 1 space, the minute-hand moves over 12 spaces 
and the second-hand moves over 720 spaces. 



MISCELLANEOUS PROBLEMS 91 

1. Prom T to h - 1, Tto M = 12, Tto 8 = 720, A to # 
= 708, tfto A = 708, h to if = 11. Hence the number of 
spaces around the face of the clock = 708 + 708 + H ~ 
1427, and the second-hand has passed over T 7 T 2 2° T of the dis- 
tance, and the time equals T 7 T 2 2° T X 60 = 30JL 9 g\ seconds 
after 12 o'clock, and it will be equally distant between the 
two hands again at ^^ X 60 = 60^^- seconds aft3r 12 
o'clock. This last answer is the one Greenleaf gave. 

2. In figure 2, T to h = 1, T to M = 12, and once around 
plus T to S = 720 spaces ; li to M = 11, M to 8^ 11; 
hence the number of spaces around the face of the clock is 
697. Therefore the time = ffi% X 60 = 61ff| seconds 
after 12 o'clock. 

3. In figure 3, 7 7 to h = 1, T to M = 12, h to if = 11, 
TtoS = 720, counting around the face, T to li = 11; hence 
the spaces once around the face = 720 + 10 = 730, and 
the time = |f$ X 60 = 59|f seconds after 12 o'clock. 

18. From a cask containing 10 gallons of wine, a servant 
draws off a gallon each day for twenty days, each time 
supplying the deficiency by the addition of a gallon of water; 
and then, to escape detection, he again draws off 20 gallons, 
supplying the deficiency each time by a gallon of wine. 
How much water still remains in the cask ? 

Solution. After each day's drawing from the cask r 9 ^ of 
its previous contents will be left. The quantity of wine 

left the first day is T 9 ¥ of 10 gallons ; second day, — - % of 10 

920 
gallons; and at the close of the twentieth day = --^5 X 10. 

Now, the quantity of water in the cask at the end of the sec- 

/ 9 20 
ond twenty days would be equal to the quantity f — ^ X 10 

920 
multiplied by j^. 

.9 20 = . 12157665459; 



92 ABITHMETIC. 

and 

10 - .12157665459 x 10 = 8.7842334541 gallons, 

the quantity of water in the cask at the end of twenty 
days; hence 

8.7842334541 X .9 20 = 1.0679577 
gallons of water still in the cask. 

19. James Page has a circular garden, 10 rods in diam- 
eter; how many trees can he set in it so that no two shall be 
within 10 feet of each other, and no tree within 2\ feet 
of the fence inclosing the garden ? 

Solution. Deduct 5 feet from the diameter and it leaves 
160 feet. Planting 1 tree at the center of the circle, next 
set 6 other trees around it 10 feet from each other and from 
the center tree. These 6 trees form the six corners of a 
regular hexagon. Now plant another hexagonal row of trees 
around the first row at the required distance and it will 
contain 18 trees, 4 trees in each side of the second hexagon. 
Repeating this process, eight rows can be set around the 
center tree, until the eighth row contains 48 trees. Since 
the first hexagon contains 6 trees and the eighth 48 trees, 
the total number = (6 + 4.8) X 4 = 216. 

Counting the tree at the center, we have 217. But the 
sides of the eighth hexagon do not occupy all the space that 
can be used. Between each side and the circumference 
there is room for 4 more trees, or 24 trees additional ; there- 
fore the total number of trees is 217 + 24 = 241. 

Remark. This problem is easily solved by planting 17 
trees on the diameter passing through the center of the 
garden, and then on the chords V 75 feet apart, parallel 
to the first diameter. The trees are planted in the quin- 
cunx order. 



MISCELLANEOUS PROBLEMS. 93 

20. A, B, C, D, and E play together on this condition: 
that he who loses shall give to all the rest as much as they 
already have. First A loses, then B, then C, then D, and 
last also E. All lose in turn, and yet, at the end of the 
fifth game, they have all the same sum, viz., each $32. 
How much had each before they began to play? 

Solution. The easiest way is to reverse the work, as fol- 

E 

$32, what each had when they quit. 
96, end of fourth game. 
48, end of third game. 
24, end of second game. 
12, end of first game. 
6. Answer. 

21. Seven men purchase a grindstone of 60 inches in 
diameter. What part of the diameter must each grind off 
so as to have one seventh of the whole stone ? 

Solution. There remains six sevenths of the stone after 
the first has ground off his part. 

1st diameter = 60 ff = «yi V±2 = 55.54921 inches ; 



lows : 








A 


B 


C 


D 


$32 


$32 


632 


$32 


16 


16 


16 


16 


8 


8 


8 


88 


4 


4 


84 


44 


2 


82 


42 


22 


81 


41 


2L 


11 



2d 

3d 

4th 

5th 

6th 



= 60 V\ = %jl V35 = 50.70925 
= 60 ff = \°- V28 = 45.35574 
= 60 V% = -yi |/21 = 39.27922 



= 60 V\ = -V- 1 7 14 = 32.07135 
= 60 V\ = \°- V'Y = 22.67787 
Subtracting the first from 60, and the second from the 
first, and so on, the answers are 4.45079, 4.83996, 5.35351, 
6.07652, 9.39348, and 22.67787 inches respectively. 

22. Four ladies purchased a ball of exceedingly fine 
thread, 3 inches in diameter. What portion of the diame- 
ter must each wind off so as to share of the thread equally ? 



94 ARITHMETIC. 

Solution. Three fourths of the ball remained after the 
first unwound her share. 

1st diameter = 3 ff = f H= 2.72568 inches; 
2d " =3 V =1^4 = 2.38110 " 

3d " = 3^ =£#2" =1.88988 " 

Hence the 

First wound off 3.00000 - 2.72568 = 0.27432 inches; 

Second " " 2.72568 - 2.38110 = 0.34458 " 

Third " " 2.38110-1.88988 = 0.49122 " 

Fourth " " 1.88988 = 1.88988 

23. Find what each of the four persons, A, B, 0, and D, 
are worth, by knowing — 

1st. That A's money together with \ of B's, C's, and D's 
is equal to $137. 

2d. That B's money together with \ of A's, C's, and D's 
is equal to $137. 

3d. That C's money together with \ of A's, B's, and D's 
is equal to $137. 

4th. That D's money together with \ of A's, B's, and 
C's is equal to $137. 
Solution. 

i (B's + C's + D's) = f (B's + C's + D's + A's) - 1 A's; 
that is, 

A's + \ (A's + B's + C's -f D's) - \ A's = $137; 
hence 

f A's = $137 - | of the sum of all. 
Or, 

A's = | of $137 — I of the sum of all. 

B's = $ of $137 - i of the sum of all. 

C's = f of $137 - i of the sum of all. 

D's = \ of $137 - | of the sum of all. 

Adding the values of A, B, C, and D, we have the sum 

of all = 



MISCELLANEOUS PROBLEMS 95 

(f + i + f + I) of $137 - (i + i + \ + i) of the sum of all 

Or, 

(l + i+i+i + i)of^mo/«K = (|+i + i+|)of$l*r; 

whence the s?m 0/ aW = 

1+2 + 3 + 4+ T 

Substituting this value of the sum of all, 

A = $47, B = $77, C = $92, D = $101. 

24. Suppose from an acorn there shoots up a single stalk, 
at the end of the year; that, at the end of each year there- 
after, this stalk puts forth as many new branches as it is 
years old; also suppose all the branches to follow the same 
law, that is, to produce as many new branches as they are 
years old. How many branches will this oak tree consist 
of at the end of twenty years ? 

Solution. At the end of the first year there will be one 
stalk, which denote by 1 ; at the end of the second year, 
1 + lo; at the end of the third year the branches, 1 -\- 1 , 
will become 1 2 + 1 ; the first, being two years of age, will pro- 
duce two new branches, the other will produce one new one; 
these can be represented by 1 2 + l x + 3 , the small figures 
denoting the age in years of the stalks or branches. The 
. results are summarized thus: 

End of 1st year, 1 = 1. 

2d " li + lo = 2. 

3d " l 2 + li + 3 = 5. 

4th * i 3 + i 2 + 3l + 8 =13. 

5th " i i + i 3 +3 2 + 8 1 + 21 = 34. 

6th " 1 5 + 1, + 3 3 + 8 2 + 21 x + 55 = 89. 

The law of the series is, that twice any term, increased by 
the sum of all the preceding terms, gives the next term ; 
or three times any term, diminished by the next preceding 
term, will give the next term. 



96 



ARITHMETIC. 



The terms are easily found by continual additions ac- 
cording to the law of the series. 

0. . . .New branches 1st year. 
Total branches 1st year 

2d " 



1 

1... 

2 

3... 

3d " 5 

8 .. 

4th " 13 

21... 

5th " 34 

55 .. 

6th " 89 

144... 

7th 4< 233 

377... 

8th " 610 

987... 

9th " 1,597 

2,584... 

10th " 4,181 

6,765... 

11th " 10,946 

17,711... 

12th ■• 28,657 

46,368... 

13th " : 75,025 

121,393 .. 

14th '■ 196.418 

317,811... 

15th " 514,229 

832,040. . . 

16th " 1,346,269 

2,178,309... 

17th " 3,524,578 

5,702.887. . . . 

18th " 9.227.465 

14,930,352... 

19th " 24.157,817 

39,088,169. 
20tb fl 63,245,986. Answer. 



2d " 

3d " 

4th " 

5th " 

6th " 

7th " 

8th " 

9th " 

10th " 

11th " 

12th " 

13th " 

14th " 

15th " 

16th " 

17th " 

18th " 

19th " 

20th M 



PROBLEMS. 97 

25. The roof of a building with perpendicular front 
makes with the horizon an angle of 45°. A leaden ball 
rolled from the apex thereof strikes the horizontal plane 
below 40 feet from the base of the front; but when rolled 
from the center of the roof it strikes only 30 feet from the 
base. Kequired the height of the front and the length of 
the roof. (Porter's Arithmetic.) 

Solution by Professor E. B. Seitz. Let AB represent 
the roof, BD the front, C the center of AB, E the point 
at which the ball strikes the horizontal plane when rolled 
from A, and F the point at which it strikes when rolled 
from C. Draw EG and FH perpendicular to DE, and 
meeting AB produced in G and H ; and draw GKL and 
HI perpendicular to BD. 

Because the angle BGL = 4b°, BL=LG = DE=±0 
feet, BI= HI=:_DF=30 feet, HE = GE = EF=10 
feet, BG = 40 V2, BH = 30 V2. The velocity attained 
by a body falling freely or down an inclined plane varies 
as the square root of the distance described; hence the 
velocity acquired in rolling down AB : the velocity ac- 
quired in rolling down CB : : V2 : 1. But with the veloc- 
ities acquired in rolling down AB and CB the ball would 
describe, respectively, the distances BG and BE, and in 
the same times that it would fall through the distances GE 
and HF under the influence of gravity; hence, since BH 
= f BG, and the velocities are to each other as V2 and 1, 
the time of describing BE is equal to f V2 times the 
time of describing BG, or the time of falling through 

HF is equal to f V2 times the time of falling through 
GE. 

Assume 1 second to be the time of falling through GE; 
then_we would have GE = 1§^ f eet = \g, HF=\gx 
(f V2f = ^g, and HE = HF - GE =^g. But HE = 
10 feet; hence the distance a falling body describes varies 



98 ARITHMETIC. 

as the square of the time; the time of falling through GE = 
VlO -v- T V<7 = 4( VlO -^ #) sec. ; .\ GE = \g X 160 -r- g = 
SO feet, and BD = BL+ GE = 120 feet. 

The velocity acquired in rolling down AB =BG + 
4 f'lO -r-# =2 1^5//; hence, since the velocity acquired in 1 
second is lg V2, we have AB = (2 V(2^)) 2 -f- # ^2 = 
10 1/2 = 11.142 feet. 

26. A and B start at opposite points to skate to the 
other's starting point; distance 8 miles. A, by having the 
advantage of a uniform wind, performs his task 2\ times 
the quickest and 48 minutes the soonest. Required the 
force of the wind per minute, and the time that each is 
skating it. (Parke's Philosophy of Arithmetic.) 

Solution. Their speeds are as 5 : 2, and their difference^ 
3 = 48 minutes, once =16. Therefore A skates the dis- 
tance in 16 X 2 = 32 minutes, and B in 16 X 5 = 80 min- 
utes; consequently A 's speed = 8 -4- 32 = \ of a mile a min- 
ute, and iTs speed = 8 -f- 80 = y 1 ^ of a mile a minute; and 
the velocity of the wind = \ — ^ = ^ of a mile a minute. 

27. A man bought a farm for $6000, and agreed to pay 
principal and interest in three equal annual installments. 
What was the annual payment, interest being 6$ ? 

Solution. $6000 at 6f compound interest in three 
years will amount to $7146.096; then we have $1.00 + 
$1.06 + $1.1236 = $3.1836; whence, $7146.096 ~ $3.1836 
= $2244.658 + 

28. If 12 oxen eat up 3 J acres of pasture in 4 weeks, and 
21 oxen eat up 10 acres of like pasture in 9 w r eeks: to find 
how many oxen will eat up 24 acres in 18 weeks. 

Remark. This problem in its generalized form origi- 
nated with Sir Isaac Newton. It has appeared in many 
arithmetics and algebras since. 



PBOBLEMS. 99 

The celebrated mathematician, Dr. Artemas Martin, 

Washington, D.C., gives the following beautiful analytical 

31 
solution: In the first case an ox eats £ of -| = ^ of an 

acre, and T 5 g of the growth of that acre, in one week; in 
the second case one ox eats { of ff = y 1 -^ of an acre, and 
Jf of what grows on one acre, in one week. 

Since one ox eats the same quantity of grass in one week 
in each case, therefore Jf — T 5 g = y 2 ^ of the growth of one 
acre during one week is = T 6 ^ — y 1 -^ = T | \% of an acre; and 
T f f- T -T- y% = y 1 ^ of an acre, what grows on an acre during 
one week. 

-fa + fa °f tV = T¥ = °f au acre > the P ar ^ °f the original 
quantity on one acre which one ox eats in one week. 

-g 5 T X 18 = f = quantity of grass, in acres, one ox will eat 
in 18 weeks. 

24 + (y 1 ^ X 24 X 18) = 60 = quantity of grass, in acres, 
to be eaten from 24 acres in 18 weeks; and 60 -f- § = 36, 
the number of oxen required to eat it. 

The following is an easy method of solving all such 
problems: 

1. Suppose each ox to eat 100 pounds of grass in 1 week; 
then 12 oxen will eat 4800 pounds in 4 weeks ; and 
4800 -T- 3^ = 1440 pounds, the whole quantity, including 
the growth on 1 acre for 4 weeks. 

2. 21 oxen will eat, at the same rate, 18,900 pounds in 9 
weeks, and 18,900 -=- 10 = 1890 pounds, the whole quantity, 
including the growth on 1 acre for 9 weeks. Since there 
was the same quantity of grass on each acre at the time the 
oxen began to graze, then 1890 — 1440 = 450 pounds must 
be the growth on 1 acre for 5 weeks, and 450 -4- 5 = 90 
pounds is the growth on 1 acre in 1 week; and the original 
quantity of grass on 1 acre = 1440 — 360 = 1080 pounds. 
Consequently, 1080x24=25,920 pounds, the original 
quantity on 24 acres, and 90 X 18 X 24 = 38,880 pounds, 



100 ARITHMETIC. 

the growth on 24 acres, and for 18 weeks 25,920+38,880 
= 64,800, the total number of pounds on 24 acres in 18 

weeks. 

3. Since an ox eats 100 pounds in one week, in 18 weeks 
lie will eat 1800 pounds; therefore, 64,800 -^ 1800 = 36 
oxen, the number required. 

Remark. Any other number may be assumed as the 
number of pounds an ox eats in one week. 



Algebea. 

Brief History. 

The word Algebra is of Arabic origin, and in its original 
form it is Algabr, which by a slight change becomes 
Algebra. Literally it means the reduction of parts to the 
whole; but for present purposes it may be defined as that 
branch of mathematics which treats of the general relation 
of quantities by means of symbols. 

The history of this science is interesting and instructive, 
and like all other sciences it has been developed gradually 
from a few elementary notions. Not far from the year 350 
a.d., Diophantus, a Greek, wrote a commentary on Arith- 
metic. In this work, a mutilated copy of which was dis- 
covered about the middle of the sixteenth century at 
Rome in the Vatican library, are solved some equations of 
the first and second degrees. He simply wrote out his 
solutions, not using any of the signs now employed. 

The problems that Diophantus solved, belonging to our 
elementary Algebra, were of the following forms: 



also, 



(1) x + y = a, 

and (2) z 2 + y 2 = 5, to find x and y; 

(1) x -y =a, 

and (2) a? -f- y 2 = b, to find x and y. 



These equations lead to the simplest forms of quadratics; 
but if the solutions were written out in full without the 
aid of symbols, the processes would be quite tedious; hence 
it is not a matter of great surprise that no more progress 

101 



102 ALGEBRA. 

was made. But in another direction Diophantus achieved 
greater success, and in fact he laid the foundation for that 
interesting class of problems now called "The Indeter- 
minate Analysis," which relates chiefly to square, cube, and 
biquadrate numbers, and to rational right-angled triangles 
and other polygonal figures. 

The fragments of Algebra that have come to us from 
Diophantus show that he was a mathematician of no 
ordinary skill for that age. His solutions of indeterminate 
problems of the second degree effected by the cumbersome 
methods he employed attest his ability, and the fact that 
the science remained stationary, say, for a thousand years, 
as it came from his hands, is the strongest evidence of his 
analytical power. 

As an illustration of what Diophantus was enabled to ac- 
complish, I select a problem which he solved, namely: 

"To divide a given square number, a 2 , into two other 
squares." 

Solution. Let x 2 be one of the required squares, and 
a 2 — x 2 the other. 

Put a 2 — x 2 = (a — lx) 2 , and x — h2 . 

Substituting the value of x, we have 

The values of a and b may be assumed at pleasure. If 
a = 4, J= 3, a? = V; also, if a = 10, b = 2, x = 64. This 
solution is in the ordinary algebraic form. 

Prof. G. Gill originated a new method of solving such 
problems. Here is his solution. Given x 2 + y 2 = « 2 , so 
that if we take x — a sin A, y = a cos A; we shall have 

x 2 + y 2 = a 2 (sin 2 A + cos 2 A) = a 2 . 
A is any angle whose functions are rational numbers. 



BRIEF HISTORY. 103 

Dr. Artemas Martin solved the same problem as follows : 
Solution. Let x 2 and y 2 be the numbers; then 

7? + y 2 = □ = z 2 (1) 

This equation is satisfied by x = z sin 0, y = z cos 0. 
Take, therefore, 

cot A = — and 2 = m 2 + w 2 ; 
n 

then 

a; = 2mn and 2/ = m2 ~~ ^ 2 « 

If m = 2, n = 1, 

x = ±, y = 3, 2 =5, 

which are the least integral numbers. 

The expressions m 2 + ft 2 * w» a — »*j and 2wm represent the 
three sides of a right-angled triangle, and no doubt it was 
owing to this discovery that the problem, " to find the sides 
of a right-angled triangle in integral numbers," suggested 
itself ta mathematicians. 

Diophantus placed in his treatise the law of the minus 
sign when he announced that "minus multiplied by minus 
produces plus/' the truth of which is still accepted by most 
authors of the present without an attempt at an explanation 
satisfactory to the beginner. 

Hypatia, the celebrated daughter of Theon, wrote a com 
mentary on the work composed by Diophantus, but this 
was lost, as well as a similar treatise that she had prepared 
on conic sections. Her knowledge of philosophy and math- 
ematics caused both men and women to become jealous 
of her, and she was torn to pieces with a harrow, — some 
assert, however, with hot pincers. 

About this time Eome broke into two great empires. In 
the dissolution of a thousand years, algebra was not culti- 
vated except by the Arabs. What the people in India were 
doing, we know not. In the library of Oxford is a manu- 



104 ALGEBRA. 

script copy of an Arab Algebra bearing the date 1342. 
This is the oldest copy of an Arab Algebra there is in 
Europe. The evidence appears quite conclusive that the 
Arabs derived their knowledge of this branch from the 
Hindus. Yet recent research tends to establish the fact 
that the Hindus made little progress in the science com- 
pared to the advancement made by the modern nations of 
Europe and of this country. 

The first one to introduce Algebra, or rather to revive it, 
in Europe was Leonardo, a merchant of Pisa. He had 
traveled extensively in the East, where he picked up some 
information on the subject of Algebra, and upon his re- 
turn he taught pupils, and afterwards published two trea- 
tises which were written in Latin verse. His knowledge 
of the subject was confined to the treatment of a few 
special equations of the first degree. 

It was not till 1505 that an equation of the third degree 
was solved. Ferreus, a professor of mathematics, Bologna, 
was the first to solve this problem in special cases. In 
those days it was the custom when one had made a new 
discovery to conceal it from others, and then to frame an 
arithmetical problem involving the discovery, and send it as 
a challenge. Ferreus kept his secret nearly thirty years, 
when he communicated it to a pupil of his, a Venetian, 
Florido, who sent it as a challenge to Tartalea of Brescia. 
Tartalea had not only solved the problem proposed, but he 
had solved three other special cases. He in turn challenged 
Florido, who failed. 

Contemporary with Tartalea was Cardan, a man of great 
ingenuity and skill. Under an oath of secrecy he obtained 
Tartalea's method of solving cubic equations, and then 
published it as his own. All algebraists are familiar with 
Cardan's (Tartalea's) formula for the resolution of cubic 
equations. 

At this time it does not appear that a general solution 



BRIEF HISTORY. 105 

of a complete equation of the second degree had been 
effected. Every complete equation of this degree may be 
placed under one of the four forms: 



(1) 


x 2 -j- %p% = q\ 


(2) 


a? + %P X — — T> 


(3) 


x? — 2px = q; 


(4) 


x 2 — 2px = — q. 


These four forms a« 


3 comprehended under the more gen- 


eral equation 






x 2 ± 2px = ± q, 


and whose roots are : 




For (1) 


X——p± Vq +jt> 2 ; 


(2) 


x = p ± Vq + p 2 ; 


(3) 


x = — p ± V— q +p 2 ; 


(4) 


x—p± V— q + jA 



Forms (1), (2), (3), and (4) are included under the more 
comprehensive equation x 2 ± 2px = ± q, which had not 
yet been solved by the keenest analysts when Scipio Fer- 
reus solved the special cubic equation x* -f bx = c. He 
had found how to make the second term of a cubic equa- 
tion disappear by substitution, and thus to reduce certain 
equations to the special form above. His solution was as 
follows: 

Assume y -\- z = x, and 3yx = — b. Then substituting 
these values in x 3 + bx = c, it becomes 

y 3 + ZyH + Zyz 2 + z 3 + b(y + z) = c = 

f + z*-b(y + z) + b(y + z) = y 3 +z* = c. 

Squaring y 3 -{- z 3 = c, and subtracting four times the cube of 

yx = — - from it, we have 

y * _ 2fz 3 +z* = (? + -foP, 



106 ALGEBRA. 



or 



* , -*=V*+£ 



By addition, 



and 



and 
whence 

y + * 



= x = \/\ 



*~\+v'+%> 




-yW'+%. 




'■r^ 




.^f-vW-, 




/?+*^fnV?- 


Vj 1 4J3 



which gives one root of the equation. 

Tartalea solved the equations a? — bx = c, and a? — ■ bx 
— — c, as well as the equation a? + bx — c; but Cardan 
received, undeservedly, the credit. Cardan's formula fails 

when & + —is negative, and yet the three roots are real 

and unequal, as may be easily shown. But that particular 
form of the cubic equation in which the roots are all real is 
called the "Irreducible Case/' while the "Ecducible Case" 
is encountered when two of the roots are imaginary or are 
equal. Having obtained one value of the unknown quan- 
tity in a cubic equation, the cubic can then be depressed to 
one of the second degree, and solved as a quadratic. 



BRIEF mSTORT. 107 

Every cubic must have at least one real root; imaginary 
roots enter by pairs. 

Biquadratic Equations. 

Mathematicians next turned their attention to equations 
of the fourth degree. The newly discovered methods for 
solving cubics were not applicable to biquadratics. Some 
contended that a general solution of a complete equation of 
the fourth degree was impossible. Cardan did not think 
so, and he asked his pupil, Lewis Ferrari, a young man of 
remarkable analytic skill, to discover a solution. Ferrari 
not only solved the special problem which others were 
unable to reduce, but he likewise effected a solution of the 
general problem. He made the solution of the equation 
of the fourth degree depend upon the solution of a cubic. 
His method of reduction was the following : 
Let #* + (*>tf + bx 2 + ex + d = be a complete biquad- 
ratic equation. Assume 

(x 2 + \ax + p) 2 - (qx + rf 

= x* + ## 3 + bx 2 + ex + d. . . (1) 

Expanding, we have 

x* -\- ax? -\- (2p + ia 2 — q 2 )x 2 + (ap — 2qr)x + p 2 — r 2 

= #* + ax 3 + bx 2 + cx + d. . . (2) 

Equating like coefficients of the same powers of x> we 
have 

*P + i -S- = ».... (3) = 2p+^-b = f; 

ap — 2§r = c. . . • (4) = ap ■— c = 2qr ; 

j) 2 — r 2 = d (5) = p 2 — d — r 2 . 

Since the product of the absolute terms of (3) and (5) is 
equal to i of the square of the absolute term of (4), we 
have 



106 ALGEBRA. 

2/ + (£ - &y - 2* - *(* - *) 

= i(«y - 2acp + c 8 ), ... (6) 
or 

jy can be found under the solution of the cubic as hereto- 
fore explained. 
Also, from (3), 



and from 


(4) 


and 


? = 

(5), 
r = 


■ |/2p + 

O/J — c 

2q 


a* 




Again, 


- Vp*- 


■d. 



: ^_|_ ax* + ix 2 + cz + d = {x 2 + ^ + jt?) 2 - (gw + r) 2 = 0. 
It is evident that 

(a* + ^ + p y = (^ + r )«, ... (8) 

and 

# 2 + \ax -\- p = qx -\- r, 
or 

# 2 + (\ a — g) x = r —p. 

Substituting the above values of p, q, and r and arranging, 
we have 

& + [$<* T |/ 2p + j - h) x + P =F VjF^d = 

when op — c is positive, and 

x* + ^ a T \/2p + | 2 - ft) « + p ± Vp 2 - d 

when ap — c is negative. These two quadratics give the 
four roots of the proposed equation. 



BRIEF HISTORY, 109 

If we put^ = z + -, and substitute in (7), it becomes, 
after reduction, 

One root of this cubic can now be found, and the equa- 
tion depressed afterwards to a quadratic, when the remain- 
ing values can be ascertained. Since the equation of the 
fourth degree is made to depend upon the solution of a 
cubic, the same difficulties are encountered in regard to the 
"Irreducible Case" as in the original cubic. 

As an illustration of this method, let there be given 

x* - 10^ 3 + 35a? - bOx + 24 = 0, 

to find all its roots. 

a = — 10, b = 35, o = — 50, d = 24. 

Substituting these values in the general form of (8) and 
reducing, it becomes 

?3 _ 13 ~ — _3 

* 12* ~ 1 08 * 

which solved by the rule previously given for binomial surds, 
z — £. Solving the resulting quadratics, the roots 1, 2, 
3, 4 are obtained. 

Notwithstanding the resolution of the biquadratic equa- 
tion, the true nature of equations was not yet understood. 
Cardan himself was unable to explain the e< Irreducible 
Case," which was not explained till Bombelli, an Italian 
mathematician, published his treatise on Algebra in 1572, 
and he noted the fact that the algebraic solution of this 
"case" corresponded to the ancient problem, "how to 
trisect any angle." 

Bombelli also solved the expression how to extract the 

cube root of a + V— 5. 



110 ALGEBRA. 

The following rule, which bears his name, states his 
process: 

" First find Va 2 + b; then, by trials, search out a number, 

r, and a square, Vd, such that the sum of their squares, 

g ■ 

c 2 + d, be = y a 2 + b, and also c z — Zed be = a: then shall 

c + V — rf be the cube root of a + V — 6 sought." 

This rule will be applied in the solution of the following 

problem, namely: Extract the cube root of 9 + 25 V — 2. 

Here a = 9, b = 1250; hence Va* + b = ^81 + 1250 = 11; 
also, 6 >2 + ^ — 11> an( l 6 ' 3 — Serf = 9, which are true when 
c = 3, d — 2; therefore the cube root is 3 -J- V — 2. .4?k<?. 

Contemporary with Cardan and Tartalea were two Ger- 
man mathematicians, Stifelius and Schenbelius, who, inde- 
pendent of what the Italians had done, did much to im- 
prove algebraic notation. The symbols for addition, 
subtraction, and the square root were first employed by 
Stifelius. His chief work was published in 1544. 

Kobert Kecorde, a teacher of mathematics and doctor of 
medicine, published the first algebra in the English lan- 
guage, which he called the "Whetstone of Wit," etc. 
He introduced the sign of equality. 

Vieta, a celebrated French mathematician born 1540, 
made algebra a symbolical science, and he was the first to 
represent known quantities by general characters or letters. 
This notation at once made a special solution-general, and 
saved the trouble of going over each problem as it was 
proposed. One solution, when general characters are em- 
ployed, solves all problems of that class. The result is 
embraced in a general formula, and the numerical values 
substituted in the formula will satisfy the conditions what- 
ever the nature of the special problem. Vieta's method 
included geometrical problems as well as algebraic, and 
thus led to great improvements in other departments of 



BRIEF HISTORY. Ill 

mathematics. He was the first to solve equations by ap- 
proximation. Vieta died at the age of sixty-three. He 
printed his writings at his own expense, and liberally 
bestowed them on men of science. 

Prior to Vieta's time, Albert Girard, a Flemish mathe- 
matician, had made considerable progress in algebra beyond 
what others had done. He was the first to interpret the 
negative sign in geometrical problems, and to point out 
the nature of imaginary quantities, and to call attention to 
the fact that there are as many roots in an equation as 
there are units in its degree. 

Thomas Harriott, born at Oxford 1560, was an eminent 
mathematician in his day, and he did much toward further- 
ing the science of algebra. His most important contribu- 
tion was that every equation was compounded of as many 
equations as there are units expressing its order. For 
instance, a quadratic is composed of the product of two 
simple equations, a cubic of three, and so on. Harriott 
gave compactness to algebraic language. 

By slow steps the science of algebra was developed. 
Each discovery required time for its exploration and its 
application. Karely did it happen that the one who made 
a discovery lived to complete it. Vieta applied algebra to 
geometry, but it was left to Descartes to apply and to in- 
terpret the doctrine of curves. He expressed the relation 
of lines by the aid of algebraic symbols, which constitute 
the equation of the curve. Hallam in speaking of Des- 
cartes says: "One man, the pride of France and wonder 
of his contemporaries, was destined to flash light upon the 
labors of the analyst and point out what those symbols, so 
darkly and painfully traced, might represent and explain." 

Descartes observed also that, in any complete equation, 
when the roots are all real the number of positive roots is 
equal to the number of variations of signs, and the num- 



112 ALGEBRA. 

Imt of negative roots is equal to the number of permanences 
of signs. This is known as " Descartes' Kule." 

Newton gave to the world the " Binomial Theorem/' but 
left no rigorous demonstration of it, and at the beginning 
of the present century Lagrange, one of the most distin- 
guished of the French mathematicians and promoters of 
algebra, discovered that every numerical equation has a 
root, either real or imaginary, which, when substituted for 
the unknown quantity, will reduce the equation to zero. 
Gauss, the German analyst, in 1801 also developed the 
subject of "Binomial Equations," and in 1819 William G. 
Horner, Bath, England, published his method of solving 
numerical equations of any degree. 

Following closely upon the discovery by Horner, Sturm 
developed the beautiful theorem which bears his name, 
the object of which is to find the limits and situation of 
the real roots of an equation. Of late years so many dis- 
coveries have been brought to light in this science by able 
analysts that the reader is referred to the recent publica- 
tions of Europe and of this country for further informa- 
tion. So great have been the extensions of this science 
that the modern higher algebra may be regarded as the 
highest achievement of mathematical skill. 

On Teaching Algebra. 

For a pupil to drop arithmetic and to commence algebra 
is a leap from the known into the unknown. A new lan- 
guage and strange and unmeaning terms confront him 
upon every hand. Little of what he learned in arithmetic 
is related to anything he finds in this new field of investi- 
gation. Some terms, it is true, bear resemblances to cer- 
tain features of arithmetic, yet in such a remote manner 
that the connection cannot be traced. Heretofore he 
worked with numbers which he added, subtracted, multi- 



ON TEACHING ALGEBRA. 113 

plied, divided, raised to powers, and extracted their roots; 
now he deals with letters and signs in a dim sort of way, 
with but little light to help him. Unsteady are his steps 
at the beginning; and so slow is the process by which the 
mind, accustomed by long 'practice to run in fixed lines, 
turns from old modes of thought to the new ones, that he 
spends much of his time in trying to adjust himself to his 
strange surroundings. The chief difficulty for him is to 
think in algebraic language by the extension of arithmeti- 
cal nomenclature; but if the teacher himself is skillful in 
closely connecting the two sciences and in showing how 
the processes of the one are related to the processes of the 
other and helps to explain them, the pupil's progress is 
easy and rapid. The unrelated presentation of algebra is 
the chief cause of perplexity and discouragement to the 
pupil. 

There are two distinct methods, or plans, teachers pur- 
sue in leading their pupils in an attack on algebra. One 
class of teachers manifest great haste in putting their 
pupils to work and devote little time to definitions and 
principles. Pupils under this system of tuition frequently 
work through an elementary algebra without being able 
to give one correct definition, and their knowledge of what 
they have gone over is vague and unsatisfactory. 

The other method assumes that when a pupil is properly 
prepared to begin algebra, he has reached such a degree' 
of mental development as will enable him to master the* 
subject easily. Maturity of mind is required, and precise 
and clear-cut thinking is demanded. Algebra is not to be 
studied in order to throw light on the shadier portions 
of arithmetic, but as an enlarged or unabridged edition 
of arithmetic. 

As a science, then, arithmetic should have been thor- 
oughly mastered, and especially mental arithmetic, before 
the pupil is permitted to commence algebra. 



Hi ALGEBBA. 

In beginning algebra, the preliminary definitions should 
first occupy the pupil's attention. Definitions are there- 
fore to be learned, explained, and understood. Definitions 
are " hitching-posts," and the learner or teacher who has 
his definitions ready to hand is doubly equipped for attack or 
defense. Besides, there is a strong educational value, as 
well as a logical value, attaching to exact definitions. Neither 
is it supposed that in learning a definition its full import 
is apprehended by the learner at once ; but rather as his 
mental horizon of the subject enlarges, he refers backward 
to the definition and he sees that it still holds true. Were 
the mind so constituted that by learning definitions the 
whole subject would instantly unfold itself, then there 
would be nothing for the mind to do except to learn the 
definitions and not the subject. Definitions are the broad- 
est generalizations of the human mind, and the strength, 
the beauty, the elegance, and the universality of a good 
definition depend upon its perfect adaptability to every 
stage of the learner's knowledge of the subject. And here 
I will remark that correct mathematical definitions are 
less variable than any other class of definitions in other 
departments of learning. 

In teaching algebra it is maintained that the true test of 
successful teaching is measured by the knowledge of the 
entire class, rather than by the achievements of the very 
few, or perhaps of one member only. 

In the remainder of this paper I will emphasize the top- 
ics which appear to me as the most important for the pupil 
to know in order to insure his progress. 

What does a definition, an explanation, a solution, or a 
principle mean to a pupil ? How does the matter, what- 
ever it may be, appear to him, and how does he under- 
stand it ? It is the pupil's conception of the subject that 
must be sought for, not the teacher's. 



ON TEACHING ALGEBRA. 115 

The following should be understanding^ learned by the 
pupil : 

1. Algebra is that branch of mathematics in which the 
relations of quantities are expressed and investigated, and 
the reasoning abridged and generalized by means of figures, 
letters, and signs. 

2. Figures, letters, and signs are called symbols. 

3. Symbols are divided into three classes : 1. Symbols 
of Quantity; 2. Symbols of Operation; 3. Symbols of Ee- 
lation. 

4. Symbols of Quantity are figures and letters. Figures 
and the first letters of our alphabet denote known quanti- 
ties; and unknoivn quantities are denoted by the final let- 
ters of our alphabet. Sometimes Greek letters are em- 
ployed to denote either known or unknown quantities. 

Zero (0) denotes the absence of quantity, while oo is the 
symbol called infinity, or a quantity without limit. Ac- 
cents and subscripts are attached to symmetrical quanti- 
ties; thus, «j a", a'" } and a v « a , a % , etc. 

5. The Symbols of Operation are -\-, — , X or [.], -^-, 

I \ • //2 /y3 r n ,~ ill 

L > 9 } 9 • 9 u 9 U 9 • • • *? 9 V } <z> 3> 4> * * • n' 

6. The Symbols of Relation are =,>,<,:,::: : , oc, 

I 9 """ 9 \)9 L J' 1 \> 9 •*• 9 '•" 

The meaning of the symbols must be thoroughly mas- 
tered at the outset. They are the mathematical alphabet, 
and progress is impossible without them. 

7. The pupils must learn the meaning and the applica- 
tion of the following : 1. Coefficient ; 2. Exponent ; 3. 
Power ; 4. Root ; 5. Reciprocal ; 6. Positive Term ; 7. 
Negative Term ; 8. Simple Term ; 9. Compound Term ; 
10. Similar Terms ; 11. Dissimilar Terms ; 12. Homoge- 
neous Terms ; 13. Monomial ; 14. Binomial Trinomial ; 
15. Polynomial ; 16. The Degree of a Term. 

The technique of the science should be mastered at once. 
Exact knowledge at the beginning is the true motto. 



110 ALGEBBA. 



Suggestions. 

1. What is the difference between the coefficient and the 
exponent of 3:s 3 ? Which is the exponent? Which the 
coefficient ? 

2. Compare ^a 2 bx?y^ and — 9ab 2 xiy 2 , and also compare 
each with 6a 2 bcx?y 2 + 9b 2 c 2 x^y^ by indicating their agree- 
ments and differences as referred to the preceding defini- 
tions. Let no confusion of thought linger in regard to 
the difference between "coefficient" and "exponent," 
"similar" and "dissimilar" terms, and "positive" and 
"negative" signs. 

Teach everything for all time. 

8. In teaching Addition, Subtraction, Multiplication, 
and Division, let the pupil compare each process with the 
corresponding process in arithmetic, thus: 

1. Add ax, bx, —ex, dx. 

2. Add 21, 15, -8+6. 

What are the agreements in the two processes ? What 
are the differences? Does the algebraic process include 
anything not contained in the arithmetical process? What 
is it? 

3. In what respects do Addition and Subtraction in 
algebra differ ? Why are they not the same ? What is the 
difference in regard to the use of the positive and negative 
signs in Addition and in Subtraction ? How are like terms 
added or subtracted? How are their coefficients affected ? 
How their exponents ? Give the law of the signs in Addi- 
tion ; in Subtraction. 

4. Removal or Introduction of Parentheses. 1. When 
the expression is preceded hy ])lits ; 2. When the expres- 
sion is preceded by minus; 3. When expressions occur 
with more than one parenthesis, or parentheses within a 
parenthesis ; 4. To put any number of terms within a 



ON TEACHING ALGEBRA, 117 

parenthesis ; 5. To put any number of terms within a pa- 
renthesis, and the minus sign placed before it. 

5. To translate algebraic expressions into ordinary lan- 
guage, and to translate ordinary language into algebraic 
expressions, thus : 

1. Find the sum of 

3{a 2 - byf - 2 Va(a -~by) + 7(a— hy)% 9 

and write the result in ordinary language. 

2. If a equals 3, b equals 2, y equals 2|, what is the nu- 
merical value of the expression ? 

3. If y equals 0, and the values of a and b are unchanged, 
what is the value ? 

4. If a equals \, b equals |, y equals \, find the value. 

9. Multiplication and Division. I. In Multiplication 
the pupil must look out for the following points: 1. The 
coefficients, numerical and literal, and their signs, as they 
occur in the partial or in the total product ; 2. The ar- 
rangement of the literal factors ; 3. The exponents of the 
quantities ; 4. Contrast the law of the exponents when like 
literal factors are employed in Multiplication with the law 
in Addition and in Subtraction ; 5. Show the difference in 
Division in regard to the exponent in the quotient as com- 
pared with the exponent in the product in Multiplication. 
6. What does Division include that is not found in Multi- 
plication ? 7. Is there any similarity in pointing decimals 
in Arithmetic and the laws of the exponents in the four 
fundamental rules of Algebra ? 8. How do they agree ? 
9. How do they differ? 

II. The Laivs of the Signs. 1. Like sign multiplied by 
like sign ; 2. Sign multiplied by unlike sign ; 3. Like sign 
divided by like sign; 4. Sign divided by unlike sign. 5. 
The Generalized Laws are : (1) Like signs give plus ; (2) 
Unlike signs give minus ; (3) If the number of minus fac- 
tors of a product be odd, the product is odd ; (4) An odd 



118 ALGEBRA. 

power of a negative quantity is minus ; (5) The effect of 
changing the sign of a factor, or of changing the sign of 
the dividend or of the divisor. 

The pupil should make haste slowly, but surely, in Addi- 
tion, Subtraction, Multiplication, and Division. Every- 
thing ought to be thoroughly understood by him except 
the full force of the definition of Algebra, which he only 
partially apprehends now, before he takes up additional 
topics. Frequent reviews, references, cross-references, 
comparisons, and entire familiarity with algebraic nota- 
tion, are absolutely required. The subject must be cleared 
of all doubts and difficulties. There must be no uncer- 
tainty in the pupil's mind. Each principle, law, opera- 
tion, deduction, must be firmly grasped and held in the 
mind, ready for instant use. 

Problems graduated according to complexity and diffi- 
culty help to develop original power. One problem 
changed as to some one of its conditions each time will give 
a better view of a subject than many different problems. 
One new step at a time is best for the average mind. Prog- 
ress is measured by the pupil's skill in mastering original 
problems. Judicious practice that stimulates but does not 
discourage is the sure road to success. 

Theorems. 

10. The theorems are employed chiefly to contract work. 
It is more convenient to work with small expressions than 
with complicated ones. Certain forms occur so frequently 
that the pupil should learn them once for all. The chief 
elementary theorems should be known and remembered. 
Their importance cannot be overestimated. 

Theorem 1. The square of the sum of two quantities is 
equal to the sum of their squares plus twice their product. 
Thus, 

(20 + 5) (20 + 5) = 400 + 25 + 200 = 625 ; 



THEOREMS. 119 

or 

(a + i) ( a + b) = (a + §) 2 = a 2 + & 2 + 2a& = a 2 + 2a5 + 6 2 . 

Theorem 2. The square of the difference of two quanti- 
ties is equal to the sum of their squares minus twice their 
product. Thus, 

(20 - 5) (20 - 5) = 400 + 25 - 200 = 225 ; 
or 
(a - b (a - b) = (a - b) 2 = a 2 +b 2 - 2ab = a 2 - 2ab + b\ 

Theorems 1 and 2 are symbolically expressed thus: 

{a ± b) 2 = a 2 ± 2ab + b\ 

Theorem 3. The product of the sum and difference of 
two quantities is equal to the difference of their squares. 
Thus, 

(20 + 5) (20 - 5) = 400 + 100 - 100 - 25 = 400 - 25; 

or 

(a + b) (a-b)=a 2 + ab- ab -b 2 = a 2 - b 2 . 

Theorem 4. The difference of the same powers of two 
quantities is divisible by the difference of the quantities. 
Thus/ 

(a 2 -b 2 ) ~ (a-b) = a + b. 

( a 3 _ JS) _^ ( fl _ £) = a 2 + ab _j_ b 2 % 

(a 4 - b 4 ) -v- (a - b) = a 3 ■+ a 2 b + aZ> 2 + S 3 = {a 2 + S 2 ) (a + b). 

(^ - y) -*- ( a - J) = a* + a 3 6 + a 2 b 2 + aS 3 + bK 

The pupil will observe the law by which the exponent of 
the leading letter decreases, while the exponent of the 
other increases ; also, that the sum of the exponents in the 
same expansion is invariable, and that the number of terms 
in each expansion is equal to the number of units in the 
exponent. 



120 ALGEBRA. 

THEOREM 5. The difference of the same even powers of 
two quantities is divisible by the sum of the quantities. 
Thus, 

(a* - b 2 ) -=- (a + b) = a -b. 

(V - b') -r-(a + b) = a 3 - a 2 b + ab 2 - b\ 



(a 6 - b«) 



( a + b) = a 6 - a*b + aW— a 2 P + aV - b\ 



Theorems 4 and 5 may be extended when the exponents 
in the dividend are different, if the same difference exists 
in the divisor. Thus, 

( a 9 _ tf) + (^3 _ J2) ^ ^6 + rftf _|_ fc 

(flS_ J12) + j^2 + £3) = a 6 _ ^£3 + ^6_ £9 # 

The law in regard to the exponents can be extended to 
coefficients, if the coefficients of the dividend are the same 
as the corresponding terms of the divisor. Thus, 

(256a 12 - 81b s ) -f- (4a 3 +36 2 ) = 64a 9 - 48a 6 £ 2 + 36a 3 &±-275 6 . 

Theorem 6. The sum of the same odd powers of two 
quantities is divisible by the sum of the quantities. Thus, 
( a * + jsj _^_ ( a + $) _ a 2 _ ah + Vm 

( a 5 + J5) + ( a + J) = ^ _ fl SJ + ^2£2 _ aV + J4 # 

Also, 

( a 12 + J9) _^_ (^ + J3) = a % __ tftf + J6 # 

In the case of coefficients the extension is the same as 
in Theorem 5. 

Let the pupil compare Theorems 4, 5, and 6. Drill 
upon these theorems till they are well understood. 

Theorem 7. The product of two binomials whose first 
terms are the same is equal to the square of the first term, 
plus the sum of the second terms into the first, plus the 
algebraic product of the second terms. 

Thus: 

z 2 + \\x + 30 = (x+ 5) (x + 6). 



THEOREMS. 121 

Here x is the first term of both binomials, the sum of the 
second terms is 11, and their product is 30. This theorem 
lays the foundation for the solution of all numerical equa- 
tions above the first degree. 

Again, suppose we have x 2 — Sx — 20 to be factored. 
We must find two numbers whose sum is — 8 and whose 
product is — 20. Evidently 

_ 10 + 2 = — 8, and - 10 X 2 = - 20, 

whence 

x 2 - Sx - 20 = (x + 2) (x- 10); 

or regarding the expression as a quadratic equation, we 
have 

x 2 - 8x - 20 = (x + 2) (x - 10) = ; 
whence each factor equal zero, or 

x = — 2, and x = 10. 

Drill the pupil on exercises under this theorem. Select 
the exercises from the text he studies. Such a direction 
given to his work will save time frequently in solving equa- 
tions by other and more formidable methods. Factoring 
is "the short-cut" in the solution of problems, and skill 
in handling equations depends almost entirely upon the 
operator's tact in discovering and in suppressing common 
factors. In short, it is the ability to discover latent forms 
and to reject them while preserving the value of the 
expression. 

But it would be wrong for the pupil to conclude that all 
trinomials are factorable. The forms that can be resolved 
are : 

(1) x 2 + ax + b; (3) t? + ax - i; 

(2) x 2 — ax + i'y (-1) x? — ax — h. 



122 ALGEBBA. 

In (1) and (2) a is the sum of two quantities whose pro- 
duct is b; and in (3) and (4) a is the difference of two 
quantities whose product is b. The sum and difference are 
here used in their arithmetical sense. 

Greatest Common Divisor and Least Common 
Multiple. 

11. The Greatest Common Divisor affords a test for ascer- 
taining whether two or more quantities, however complicated, 
are prime to each other. When it is desirable to suppress 
a common factor that exists in two or more algebraic ex- 
pressions, or to search for the existence of such a factor, 
recourse is had to that process known as finding the 
Greatest Common Divisor of the expressions. As a process 
the pupil should know it. All methods designed to abridge 
or to simplify algebraic expressions ought to be well under- 
stood by the pupil. The groundwork of Algebra must be 
laid in a thorough mastery of the elementary processes of 
reduction and simplification. These are the "clearing-up 
processes" for the successful handling of equations. 

In one respect the Least Common Multiple is different 
from the Greatest Common Divisor. It is the least con- 
tainer of two or more quantities, and its use is to reduce 
dissimilar expressions, chiefly, to similar ones, that they 
may be more closely united, separated, or partitioned. 

The drills in Greatest Common Divisor and Least Com- 
mon Multiple help to fix the four fundamental operations 
more clearly in the mind ; yet the reasons for learning 
them will not so readily suggest themselves to the pupil 
unless constant reference is made to the same operations in 
Arithmetic ; but when the pupil is reminded that dissimi- 
lar fractions must be changed to the same unit of value 
before they can be added, subtracted, or divided, — in 
short, handled with any degree of satisfaction, the way is 
partially blazed out for him to trace the connection in the 



FBACTIONS. 123 

two branches. Lest the pupil mistake the nature of 
Factors, Divisors, and Multiples, he is requested to group 
them under the heading "Labor- Saving Methods." 

Let the pupil draw a sharp, hard line between how to 
find the Greatest Common Divisor of two or more quanti- 
ties and how to find the Least Common Multiple of the 
same quantities. In teaching, I regard this as better than 
taking a different exercise for each. The contrast is more 
striking. Contrasts as well as analogies are helps in setting 
matters in the mind. 

The teacher must not forget to review frequently, and to 
have the pupil trace every step back to the definitions or 
principles upon which it depends, This connects the work 
and gives the pupil confidence in himself, and it cultivates 
the habit of demanding a reason for everything he does. 

Fractions. 

Many persons fail to get a clear conception of the full 
force of the sign of a fraction ; but if the use of the " pa- 
renthesis" has been taught, little difficulty will be experi- 
enced. In looking at an algebraic fraction, the learner 
should observe four things : 

1. The sign that precedes the line between the numerator 
and the denominator. This sign is called the apparent 
sign of the fraction. 

2. The sign of the numerator, or of each term of the 
numerator. 

3. The sign of the denominator, or of each term of the 
denominator. 

4. The sign of the quotient, or of each term of the 
quotient. 

The dividing line answers the purpose of a parenthesis 
or vinculum, and when the sign that precedes the dividing 



124 ALGEBBA. 

line is changed, the sign of the fraction is changed 
also. Thus : 

a b z , ab 

— = o: but = — b: 

a a 9 

or, 

ab — ac , , . db — ac 

= b — c: but = — o + £. 

a 

Again, 



or, 



— ab 



b; but 



-ft; or 



a 




—ab__ 


b; 


a 


ab 


= b. 



— a 

Two signs when changed do not change the sign of the 
fraction. But when all three signs are changed, the sign 
is changed. Thus: 

ab , - — ab 7 ab — ab ab 

= b ; but = — b = — 



a —a a a — a 

Which shows that to change all the signs is the same in 
effect as changing one sign. That is, to change the sign 
preceding the dividing line is the same in"effect as chang- 
ing the sign of each term in the numerator and in the de- 
nominator of the fraction ; and this holds true notwith- 
standing the number of terms in either the numerator or 
the denominator. Therefore the pupil should be deeply 
impressed with this principle, that to change the sign of a 
fraction from plus to minus, or from minus to plus, changes 
the sign of each term. Frequently it is very desirable to 
change algebraic expressions from plus to minus, or from 
minus to plus, that they may be more easily dealt with. 

There is, however, another phase of this subject that is 
more difficult for the learner to understand than handling 
terms united by plus or separated by minus, and that is 



FBACTIONS. 125 

when the terms of a fraction are composed of any number 
of factors. Thus: 



(x - y)(x - z) (y -x)(z- x) ~(y - x)(x - z) 

b — a 
(x - y)(z - x) 
But it will be observed that in each case the signs of two 
factors have been changed. If the signs of all be changed 
thus, 

b — a 
lj-.x)(z -*)* 

the value of the fraction is changed. Hence the law is, 
when a fraction is composed of any number of factors, 
that any even number of factors may have their signs 
changed without changing the value of the fraction ; but 
if any odd number of factors have their signs changed, 
the value of the fraction is changed. 

To illustrate this law the following is selected : 
1 1,1 



+ 



■ ? 



a(a — b)(a — c) b(b — a)(b — c) c(c —a)(c — b) 

Solution. On account of symmetry it is preferable to 
make the common denominator abc(a — b)(a — c)(b — c). 

Hence we must change the sign of b — a in the second 
term, and also the signs of c — a and of c — b in the third 
term. But we cannot change the sign of b — a in the 
second term unless the sign of the numerator be changed. 
Hence we have . 

1 bc(b - c) 



a(a- 


-b)(a- 
1 


-c)~ 


abc(a — b)(a - 
- 1 


- c)(b - 


— ac(a 


-o) 




l{l- 


-a)(b- 
1 


-c) 


b(a- 


-b){b-c) - 

1 


~~ abc(a 


-b)(a 
ab{a — 


-c)(b- 
b) 


-cY 


c(c - 


- a){c - 


-b)- 


c(a — 


c)(b-c) - 


dbc{a 


-b)(a- 


-c)(b- 


-«)■ 



126 ALGEBRA. 

Adding the numerators, their sum is the same as 
(a — b)(a — c)(b — 6*); omitting this common factor, the 

result is — =-. 
abc 

The main point is to drill the pupil well in the law of 
the signs first; then the second important step is in the 
reduction and interpretation of fractional expressions. 
The laws of reduction are the same in principle as those in 
arithmetic; but the distinction between number and quan- 
tity ought to be preserved intact. Dissimilar algebraic ex- 
pressions must frequently be reduced to equivalent, similar 
expressions, and in making reductions the learner should 
always keep in view the fact that it saves labor to change 
complex forms into simpler ones. 

To fix principles in the mind, well-graded exercises must 
be chosen, and if the learner solves a problem in his own 
way and understands it thoroughly, and the teacher has a 
simpler solution, let him give it so that the pupil can see 
how it is solved. Let the teacher impress upon the pupil 
the similarity of the operations in arithmetic and algebra. 
Such teaching connects the two branches, and the one helps 
to explain the other. 

Here, too, is a favorable opportunity for explaining the 
limitations of arithmetic. 

In arithmetic, % equals 0, $ equals 0, and -§- equals 0. 
But in algebra the language is more general, and is used 
in two senses: first, it may denote the absence of any value 
whatever, and it is then absolute zero; second, it is most 
frequently regarded as a very small quantity, and in this 
case it is said to be an infinitesimal, which signifies a quan- 
tity approaching zero, but zero is the limit which it never 
quite reaches. Since a quantity may be infinitely small, or 
infinitesimal, likewise a quantity may be infinitely great, 
and then it is called by writers infinity. Arithmetic, then, 
is limited by zero and plus infinity; while algebra deals 



FRACTIONS. 127 

with negative and positive quantities running from plus 
infinity to minus infinity. An infinitesimal is represented 
by 0, and infinity by co . 

Since a fraction may have any value whatever, if the de- 
nominator remains constant while the numerator continually 
decreases till it is less than any assignable value, or the 
numerator remains constant while the denominator de- 
creases continually, we have the following equations, which 
must be translated into ordinary language: 

71/ n 

Let (1) - = any fraction; (2) -7 = 0; ( 3 ) tt = °° • 

Again, if either the numerator or the denominator remains 
constant while the other continually increases, or, if both 
decrease till each is less than any assignable value, there 
will result the following equations : 

(4)£- = 0; (6)£=? (8)<x>-co=? 

(5). ¥ = «oj (<)"- = ? 

Under these forms, (5), (6), (7), and (8) require special 
attention. (5) shows that two infinities can be equal in 
one special case; otherwise unequal, and indeterminate; 
(6) is the symbol of indetermination; likewise (7) and (8) 
are indeterminate. The pupil very often rushes to a hasty 
conclusion, and decides that each infinity, or infinitesimal, 
is as large, or small, as any other like quantity. 

There is another form that sometimes puzzles the learner, 
and owing to its importance in the higher departments of 
mathematics it is referred to here, and that is the vanish- 
ing fraction. To avoid ambiguity, the common factor 
must be suppressed in both numerator and denominator of 
a vanishing fraction before it is evaluated. In my opinion 
the teacher is not apt to err in giving too much attention 



128 ALGEBRA. 

to fractions. The teaching here ought to be clear-cut and 
thorough. 

Summarizing : 

1. Teach the definitions thoroughly. 

2. Give special attention to the signs. 

3. All processes of Reduction and Transformation must 
he thoroughly mastered. 

4. Discussion of zero and infinity as they occur in frac- 
tional forms. 

Equations of the First Degree. 

There are certain terms used in connection with the sub- 
ject of Equations that the learner ought to know. A 
knowledge of them is indispensable to his successful prog- 
ress in acquiring a clear conception of the subject. Mathe- 
matics has its language, which is precise and technical. 
Terms denote definite ideas, and their exact meaning must 
be learned as they occur to avoid confusion. To express 
this thought tersely, — the technique of the science must be 
mastered. 

Points to be Emphasised. 

1. The exact definitions of the terms. 

2. That the learner should get the exact meaning of a 
problem when he reads it, or when it is given to him for 
solution. 

3. That the learner should think out the relation or rela- 
tions existing between the known and unknown quantities, 
and express them in algebraic language. This is the thought 
process, and is called the statement of the problem. 

4. The solution of the problem. 

The ability to express the conditions of a problem in 
equations depends upon the analytic power of the learner's 
mind. That frame of mind most conducive to a satisfactory 
investigation of mathematical relations is acquired and 



EQUATIONS OF THE FIRST DEGREE. 129 

stimulated chiefly through that mental characteristic — 
abstraction. 

In thinking out the statement of a problem, the learner, 
from the very nature of the subject-matter, must concen- 
trate his thoughts upon the conditions implied in the ques- 
tion, and exclude all extraneous matter. 

Of course this act requires strong will power, and exer- 
cises of this character eminently qualify the learner for 
other operations of his reasoning faculties when applied to 
important affairs of life. Continuous thought is the differ- 
ence between a man and the man. 

After the statement of a problem is expressed, the learner 
must make up his mind how he ought to proceed to find 
the value, or values, of the unknown quantity. Sometimes 
it is much easier to state a problem than it is to solve it. 
A skillful algebraist foreseeing the difficulties likely to arise 
in resolving intricate expressions, will seek the simplest 
methods of expressing the conditions existing among the 
quantities, and the easiest way of finding their values. This 
suggestion resolves itself into two parts: 1. Judgment in 
stating a problem. 2. Skill and tact in solving it. Per- 
haps a generalized statement will include what is most im- 
portant in this particular sphere of algebraic work, — 
" sense enough to take advantage of short cuts." 

Dealing with Equations. 

The first principle is that the equation, if not already, 
should be in its simplest form; that is, freed from fractions, 
terms transposed, and other necessary reductions performed. 
To effect what is here outlined, the principle of Trans- 
position needs to be thoroughly taught. If the learner 
experiences any difficulty in understanding why a quantity 
changes its algebraic form when it is transposed from one 
member of the equation to the other, explanations by using 
figures instead of quantities generally remove all uncer- 



130 ALGEBRA. 

tainty. Children can frequently think in numbers much 
better than in symbols of quantity. The individual idea 
precedes the general notion in intellectual unfoldment. 

The steps in solving equations are in general the follow- 
ing: 

1. To clear the equation of fractions, if necessary. 

2. To transpose terms from one member of the equation to 
the other without destroying the value of the equation. 

3. Collecting unknown quantities in the first member of 
the equation, and the known quantities in the second member. 

4. Uniting similar terms. 

5. Dividing through by the coefficient of the unknown 
quantity. 

Equations of One Unknown Quantity. 

Perhaps as much or more ingenuity is required to state 
some problems involving one unknown quantity than to ex- 
press the relations existing among several unknown quanti- 
ties. At first, the problem should be a special one, whether 
it is to be solved by the learner or by the teacher who illus- 
trates for the benefit of the class. Many beginners can 
understand special or numerical problems, when the gen- 
eral ones confuse them. The mind appears impotent to 
make abrupt transitions, but gradual changes are effected 
easily. This indicates that the plan for the ordinary mind 
is, the special problem followed by the generalized problem; 
and probably this is the best method to pursue throughout 
the elementary course of study. The teacher in selecting 
problems for illustrating principles should choose simple 
ones. Principles are taught better and are fixed more 
firmly in the mind by simple illustrations than by compli- 
cated ones. A tangled skein is much harder to unwind 
than one which is free from kinks. 

The solution should always be a model of condensed 



EQUATIONS OF ONE UNKNOWN QUANTITY. 131 

neatness — clear enough to avoid all ambiguity, but suffi- 
ciently connected so as not to break the chain of reasoning 
at any point. A solution should be so plain that any one 
who understands mathematical language can read it off as 
easily as ordinary print. 

Suppose we have the following to solve: 

(1) 9x — ox = 9 + 7; 

(2) 4a; = 16; 

(3) x = 4. 

The terms are simply collected, and (2) is divided by the 
coefficient of x. 

In the problem f Ax + x = (- -J — 1, we have 

to transform, * educe, and divide through by the coefficient 
of x. 

Insist upon the pupil's marking each step in the solution. 

The teacher must introduce one difficulty only at a time. 
A child learns to walk by taking one step after another, — 
not two or three steps at once. So in teaching, one thing 
well taught, and then onward to another, keeping the 
dependence and connection in solid ranks. Straggling 
teaching is one of the banes of school-work. 

There is a tendency among pupils studying Algebra to 
pass away from the domain of Arithmetic, and to separate 
the two subjects so widely that no close connection can be 
preserved. Such a view, in my opinion, weakens both 
algebraic and arithmetical skill. As a remedy, the teacher 
should insist upon the learner's solving certain special 
problems arithmetically as well as algebraically. Algebraic 
problems solved by Arithmetic enable the learner to com- 
pare the two methods, and to decide upon the merits of 
both. The ability to do a thing in several different ways 
is a strong presumption in favor of "clear-cut knowledge." 



132 ALGEBRA. 



Two or More Unknown Quantities. 

As long as an equation contains two unknown quantities 
and there is no way of getting rid of either of these quanti- 
ties except by imposing an arbitrary value on one of them, 
the equation is indeterminate. There must be as many 
independent equations as there are unknown quantities. 
Combining two independent equations that are satisfied by 
the same values of the unknown quantities in such a manner 
as to obtain a single equation having but one unknown 
quantity, constitutes the process called Elimination. Not- 
withstanding how many simultaneous equations there may be 
in a problem, the process of reduction, by whatever method, 
is to find one equation containing one unknown quantity, 
and from which the value of this quantity can be expressed 
in known terms. Two equations must first be transformed 
into one ; three into two, and the two into one; four into 
three, three into two, and two into one, and so on. 



Elimination. 

Elimination by Addition or Subtraction is the first 
method acquired by the learner. After it is thoroughly 
understood, he is ready to begin Elimination by Com- 
parison. A good plan for him is to solve two equations by 
Addition or Subtraction, and then to solve the same by 
Comparison. Let him work both ways till he can solve 
equations by either method with equal facility. 

Next, he is to learn how to eliminate by Substitution. 
Knowing these three methods equally well, he is prepared 
to use whichever appears most convenient. Sometimes 
one method in a given case is far preferable to either of 
the others. To impress this truth, let the learner solve a 
problem by all three methods and then compare processes. 



OTHER METHODS OF ELIMINATION. 133 



The Form of Solutions. 

Mathematical studies afford excellent opportunities for 
teaching exactness in written composition. All work, 
whether on paper, slate, or blackboard, should be concise 
yet elegant in style, capital letters properly placed, and 
punctuation points correctly used. The work in all cases 
ought to be ready to set in type without any corrections 
whatever. Slovenly work denotes slipshod habits of body 
and mind. First-class work is done by first-class workers. 
To know what good work is, compare the methods of 
solving the same or similar problems in our best text-books. 

While the form of the solution is much, yet it is not the 
chief object. It signifies that the operator knows how to 
express himself in an artistic manner, and in language that 
is easily read and interpreted. To secure the best possible 
results, the learner or class must be led gradually upward 
and onward, mastering each new difficulty as it occurs. 
The teaching must be so done that the members of the 
class never lose confidence, each, in his own ability. Piling 
on too much and that which is too difficult, discourages per- 
sons of average zeal; whereas if the work is properly par- 
celled out, the working spirit is kept intact. 



Other Methods of Elimination. 

There are two other methods of Elimination which may 
be conveniently employed by those well skilled in Algebra. 
The first of these methods is that of Undetermined Multi- 
pliers. It consists in the introduction of a factor to which 
a definite value is assigned before the solution is completed. 
Many algebraists claim superior advantages for this method 
of Elimination in solving literal equations. As this method 



134 ALGEBRA. 

is not usually included in elementary works, two problems 
will be solved to illustrate it. 

Given (1) x -f- y = 15, 

(2) x — y — 1, to find x and y. 

(3) = m{\), mx + my = »il5. 

(4) = (3) - (1), x(m - 1) + y(m + 1) = rol5 - 7. 

Put m — 1 = 0, . \ m = 1. 

Substitute the value of m in (4), and it becomes 

2y = 8, and y = 4; and £ = 11. 

Instead of taking m — 1 = 0, we can use m + 1 = 0, 
and then find x 9 and afterwards y. 

Given (1) ax + by = c, 

(2) a'# + #'# = 6 1 ', to find x and y. 

(3) = m(l), max + ?#% = w?£. 

(4) = (3) - (1), (ma - a')z + (mb - b')y = mc - c'. 

a' 
Put ma — - a' = 0, . \ wi = — ; consequently (4) becomes 

(5) lT- y > =^- c; 



(0) = (5), y = a , b _ aV = - aV _ a / ft . 

Whence 

_ bc^-JVc 
a'b — ab r 

Have the learner point out the differences and the simi- 
larities in the numerators of x and y. What letters are 
permuted ? 

Should we desire to extend this method of solution to 



OTHER METHODS OE ELIMIXATIOX. 135 

three or more unknown quantities, it leads to some beau- 
tiful res 

Let it be required to solve the following equations : 

(1) A'x+B'y + C'z = B' y 

A*x - B 2 }/ - C 2 z = D\ 
(3) Ah: + Bhj - C*2 = B 3 . 

- m(l). mA'x — mB'y — mC'z = ml)' \ 
. nA*x — nB 2 y +nCh = nB 2 : 

(6) = (4) + 

( m A § - nA*)x + [mB' - nB*)y - (mC - nC 2 )z 

= mB' + nB) 2 - B 3 ; 

(7) = (€ -(3). 

(mA' - nA* - A*)z - (mB' — nB 2 - B 3 )y 

+ (mCT -f nC 2 - C 3 )z = mB' - nB 2 - B 3 . 

To cause y and : to vanish from (7), 

mB' + nB 2 - B 3 = 0, and mC" + nC 2 - C 3 = 0, 

whence 

mB' + nB 2 - B 3 

:r = —jr\ n j 3 ( s ) 

m A — n A- — A 6 v ' 

The values of m and n must be found from 

mB' -r nB 2 - B 3 = 0, and m C + n 0* - C 3 = 0, 
whence 

m ~ B'C - C'B 2 ' 

and 

B'C 3 -B 3 C 
11 - B'C 2 -B'C 

Substituting the values of m and n in (S). we have 

BB : C*-+-D-B l C—I} i BC--DB 2 C--B : BC 5 -I) 2 B ii <? 



(9) =$).* = 



A'B*C*+A*B*C+A*B Q*-ABK--A*B'C*-£»B*C' m 



136 ALGEBRA. 

Proceeding in a similar manner, the values of y and % are 
found to be 

_ A , D 2 G 3 +AW^C'+A^DC i -'AD s G i -A i I)'G^-A 3 D i C' 

(10) V - A > B 1 C Z+ A 2 B S C > +A S B < C 2_ A ' B SQ2_ A 2 B > C S -A*B*C' 

and 

A'B'D'+A'BW+A^B'B^-A'B^D'-A'B'B'-A^BW 



(11) 



A'BW^+A'B'C'+A^B'C'-A'BW-A'B C"-A^B 2 C" 



Examining these three results, the denominator is the 
same in each,— composed of three positive products, each 
product being composed of three factors ; and also three 
negative products of three factors each. The letters com- 
posing these factors are alphabetically arranged, and in the 
first product the letters have exponents corresponding to 
their order in the alphabet ; thus A'B 2 C S have 1, 2, 3, 
respectively. If we add one to each of these exponents, 
and write 1 wherever the sum of 3 + 1 equals 4, we 
shall have the exponents of the second product of the 
denominator as follows: 2, 3, 1 ; and for the third product, 
3, 1, 2. 

For the negative products, after the first is obtained, 
whose exponents are 1, 3, 2, the others are derived by 
adding 1 as in the case of the positive factors. Thus, 
2, 1, 3, and 3, 2, 1. Again, the numerator of (9) is the 
same as the denominator, if D be written for A. Hence a 
simple permutation gives the numerator of x. 

The numerator of (10) is found by writing D for B in 
the common denominator, retaining the exponents ; and 
the numerator of (11) found by writing D for (7, observing 
the same restrictions. 

The next step is to show how the common denominator 
can be obtained in an easy manner by dealing exclusively 
with the coefficients of x, y, and z. Let us write the co- 
efficients and the absolute terms in the order they occur in 
equations (1), (2), (3) ; we have — 



OTHER METHODS OF ELIMINATION. 137 



(1) 


A' 


B' 


V = D'; 


(2) 


A 3 


B* 


C* = D 2 ; 


(3) 


A 3 


B s 


C 3 = IP; 


(1) 


A' 


B' 


C = D' ; 


(2) 


A % 


B 2 


C 2 = D 2 ; 


(3) 


A 3 


B 3 


C' 3 = D 3 . 



I duplicate the equations in order to make the explanation 
perfectly clear. In the first set of equations, if we begin at 
A f and pass to B 2 and thence to C 3 obliquely downward, 
we have A'B 2 C 3 , the first product of the common denomi- 
nator. For the second, begin with A 2 , thence to B 3 , thence 
to C, we have the second product, A 2 B 3 C \ and beginning 
with A 3 , thence to B' in the second set, and thence to C 2 
in the second set, we have A S B'C 2 , the third product. 

To find the negative products, we count obliquely up- 
ward, beginning with A' in the second set of equations; 
thus, A'B*C\ Next, with A 2 , and we have A 2 B'C 3 ; and 
for the third product, A 3 B 2 C. Permuting as previously 
explained, we readily get the numerators from the common 
denominator. Or we may proceed thus : 

Write D instead of A in the original equations, and we 
have these expressions : 



(1) 


D' 


B' 


C>; 


(2) 


D* 


B 2 


0»; 


(3) 


V s 


B 3 


nz . 


(1) 


D' 


B' 


C"; 


(2) 


D 2 


B 2 


0*; 


(3) 


D 3 


B s 


C\ 



Beginning with D and moving obliquely downward, we 
have 

D'B 2 C 3 +D 2 B 3 C'+D 3 B'C 2 -D'B 3 C 2 -D 2 B'C 3 -D 3 B 2 C, 
which is the numerator of x. 



138 





ALGEBRA. 


rite j 


D instead of B, t 


(i) 


A' D' C; 


(2) 


A 2 D 2 C 2 ; 


(3) 


A 3 D 3 C 3 ; 


(1) 


A' D' C; 


(2) 


A 2 D 2 C 2 ; 


(3) 


A 3 D 3 ' C\ 



Proceeding as before, we have 
A'D 2 C Z +A 2 D Z C'+A Z D'C 2 -A'D*C 2 -A 2 D'C Z -A Z D 2 C' 

for the numerator of y. 

To find the numerator of z, write D for C, and proceed 
as above. 

In solving numerical equations particular attention must 
be given to the algebraic signs of the coefficients. 

This method of elimination is the beginning of what is 
called "Determinants" in the Higher Analysis, and its 
applications are so far-reaching that additional exercises 
are given in order to explain its uses. 

Given Ax + By = C, 

Dx -f Ey = G, to find x and y. 

By any of the usual methods of elimination, 

BG - C E __ AG- DQ 

X ~ AE-BD' and y ~ AE-BD* 

To solve by the foregoing method, we have 

A B= C, 
D E=G. 

Beginning with A, we get AE for the positive product 
of the common denominator; — DB for the negative prod- 
uct. Also for the numerator of x we have BG positive and 



OTHER METHODS OF ELIMIXATIOX. 139 

EC negative. For the numerator of y, AG positive, DC 

negative. 

Given (1) Gx + 5y = 61, 

(2) 3;/' + ±y = 38, to find x and y. 

Therefore 

61 X 4 - 38 X 5 _ . 6 X 38 - 3 X 61 _. 

x = — ; r — = 6, and ?/ = — — = o. 

6X4—3X5 J 6X4-3X0 

Again, j 3^. jl %y Z 10 f > to fin ^ » and y. 

Here 

12X-2-10X4 ( , 2X10-3X12 , 

x = — = 4, and y = — - = 1. 

2 X - 2 - 3 X 4 J 2 X- 2 -3x4 

Let it be required to find the values of x> y, and z in the 
following : 

(1) 3x + 4?/ - 2: = 10; 

(2) b.v - 2// + 3: = 16; 

(3) \x + 2tf + 2z = 22. 

Solution. 3 + 4 — 2 = 10: 

5-2 + 3 = 16; 

4 + 2 + 2 = 22; 
whence 

_10X-8X2+16X2X--g+gX4X3-10x2x3-lCx4x8-a8X-8X-a 

~3X-2X2-f 5X2X -24-4X4x3-3 X 2 X 3-5x4 X2-4X- 2 X-2 '' 

Since the denominator, — 58, is common, to find y we 
proceed as follows : 

3 + 10 -2; 
5 + 16 + 3; 

4 + 22+2; 
whence 

96 - 220 + 120 - 198 - 100 + 128 m 

y — ! ^ = 3, 

J — 58 ' 



2. 



140 



ALGEBRA. 



and 



- 132 + 100 + 256 - 96 + 80 - 440 

-58 * 



When the equations are symmetrical, the simplest solu- 
tion is to find the value of one unknown quantity and then 
permute for the values of the others. Much time and labor 
can be saved in the solutions of equations if the instructor 
will drill his classes thoroughly in handling symmetrical 
expressions. 

Eeference has already been made to that branch of 
mathematics called " Determinants." As one of the most 
valuable lessons that a teacher ever imparts is that of stimu- 
lating his pupils to higher endeavors, so a glimpse of what 
lies beyond frequently opens out a new prospect to a pupil 
and gives him a clearer perspective of what is vague and 
indistinct in his mind. The power transmitted, and a 
burning desire to learn more, are the fruits of good teach- 
ing; and he who does not leave his pupils with these traits 
of character deeply impressed, is not a success in the edu- 
cational work. Consequently, with what has just been 
given, the ambitious teacher can instruct his pupils in 
Elimination as taught in Elementary Treatises on Deter- 
minants. 



Thus 



-\ 



3 X xtll = 10 9\' to&ndxand ^ 



and 



y 





10 + 3 

9 + 2 


10 X 2 - 9 X 3 20-27 




1 + 3 

3 + 2 

1 + 10 

3+9 


1X2-3x3 2-9 
1x9-3x10 9-30 




1+ 3 
3+ 2 


1X2-9X1 2-9 



= 3. 



OTHER METHODS OF ELIMINATION. 



141 



This solution is performed in precisely the same manner 
as the preceding under this heading, but is more condensed. 
In Determinants it would be solved thus; 



x = 



10 3 
9 2 




1 + 3 
3 2 





The cross multiplication being performed mentally, the 
operation is greatly abridged. 

As a second exercise the following is selected: 

3x + 4y = 18 ) 



x = 



2x 

18 
1 



_ ^ > , to find x and y. 



II = - 



3 18 
2 1 



3 4 

2 - 1 



— = 3. 



In my own experience I found that those studying any 
branch of mathematics were always stimulated to greater 
exertion if glimpses were given them of what lay beyond 
their immediate knowledge, especially if the newer method 
contracted the labor of that the student already knew. 
True teaching is not measured so much by the amount of 
information or number of facts communicated as by the 
constant impetus given to the mind, by which it is carried 
forward ever after in pursuit of knowledge. Of late our 
Modern Algebra has been so extended, and in so many 
respects simplified, that newer methods of handling equa- 
tions should occupy a portion of the time usually devoted 
to this subject in our common and high schools. In other 
words, the teacher should be two or three sizes larger than 
the subject he teaches. 



142 ALGEBRA. 

Evolution. 

In Arithmetic the pupil learned how to extract the 
square root and the cube root of numbers, and perhaps his 
teacher gave him an insight into the working of Horner's 
Method in extracting higher numerical roots. The pupil 
saw that in Involution he was required to multiply quan- 
tities continually. Involution, then, is an act of involving 
a quantity, or shrouding it in obscurity. Evolution is just 
the reverse. It is to unroll or to unfold, and to make clear. 
Involution multiplies and obscures: evolution ?m-multiplies. 
Should I say to a friend, et I have squared or cubed any 
quantity, and this is its cube ; tell me the quantity," he 
takes the complex product, and from it finds the quantity 
required: this is evolution. 

The arithmetical treatment of this subject is quite a help 
to the learner. It is a forerunner, and clears the way. In 
fact, no great progress can be made in Algebra without a 
thorough knowledge of Involution and Evolution as treated 
of in the text-books on Arithmetic. 

Knowledge is so related that the skillful teacher will al- 
ways use the simpler kinds to unravel the more complex. 

The essential points to be emphasized are: 

1. The learner should examine the quantity carefully for 
the purpose of discovering whether it is a perfect power. 

2. It may be a perfect power of one degree but not of an- 
other degree. 

3. The algebraic sign of the entire expression must be kept 
constantly in mind. Look, think, and remember. 

4. The sign of each term. 

5. The sign of the odd roots of any quantity ; 

as, Va = a, and V— a — — a. 

6. The sign of the even roots of any quantity ; 

as, Va 2 = a or — a; that is^ ± a. 

6, — 

Again, V a 6 = ± a. 



EVOLUTION OF POLYNOMIALS. 143 

7. The eyen root of a negative quantity is impossible. 
Why? 

8. The ninth root of a quantity is equal to the mtli root 
of the nth root of that quantity. 

Thus, Pa= V Va; V~a = V Va. 

Evolution of Polynomials. 

The works on Algebra give methods of extracting roots, 
and nearly all of them are closely connected with the or- 
dinary methods taught in Arithmetic. After drilling a 
class thoroughly in the ordinary method of performing this 
work, either in Arithmetic or Algebra, I am fully satisfied 
from my own experience that the pupil or class should be 
thoroughly drilled in " Horner's Method.' 7 In every so- 
lution of numerical equations above the second degree, 
unless the root can be discovered by inspection, it is the 
shortest and easiest way to do the work; and in the solu- 
tion of numerical algebraic equations it is the only method 
that is used, unless an equation is resolved by some anti- 
quated process as a mere matter of curiosity. The excel- 
lence of Horner's Method consists in this, that it can be 
employed as easily in one degree as in another, and the 
work preceding in a solution helps what is to follow. The 
odd root can be as easily extracted as the even root. 

For instance, let it be required to extract the fifth root 
of 

32a 5 — 80a* + 80a 3 — 40a 8 + 10a — 1. 



Col. 1. 


Col. 2. 


Col. 3. 


Col. 4. 


2a 


4:X 2 


8a 3 


16a 4 


4:X 


12a 2 


■ 32a 3 


80a* 


6a 


24a 2 


80a 3 


80a± - 80a 3 -f 


8a 


40a 3 


80a 3 - 


40a 2 + 10a - 1 


2x 


40a 2 - 


10a + 1. 




loa- 


1. 







144 ALGEBRA. 

32a 5 - 80a 4 + 80a 3 - 40a 2 + 10a - 1 | 2a - 1 . 
32a- 5 • 

- 80a* + 80a 3 - 40a 2 + 10a - 1 

- 80a 2 + 80a 3 - 40a 2 + 10a - 1 



Remark. If the pupil does not know how to extract 
roots by Horner's Method, now is an excellent opportunity 
to teach him. 

While this is a special case, yet it will serve as a guide 
for any numerical equation, whether arithmetical or alge- 
braic. 

Involution and Evolution. 

There are certain notions in regard to Involution and 
Evolution that must be thoroughly fixed in the learner's 
mind at this time. He has learned thus far that quanti- 
ties can be added, subtracted, multiplied, and divided, 
and now he is ready to take two additional steps, namely, 
that quantities can be raised to any power, or have their 
roots extracted. 

The things to be critically distinguished are: 

1. The Base or Eoot. 

2. The Exponent of the Power. 

3. A Perfect Power. 

4. An Imperfect Power. 

5. The Sign of the Power. 

6. The Sign of the Eoot. 

7. The nth Power of a Product. 

8. The wth Eoot of a Power. 

9. The Coefficient of a Power. 

10. All Even Powers of a Quantity. 

11. All Odd Powers of a Quantity. 

The next subject requiring attention is the treatment of 
monomials. Since a monomial may have any number or 



POLYNOMIALS. 145 

known quantity for a coefficient, the learner must observe 
particularly the sign of the monomial, the sign of the co- 
efficient of the power, and the sign of the exponent. 

Suppose the quantity to be 2a which is to be raised to 
the ^th power. 

Then (2a) n = 2 n a n . 

If n — 2, we have 

(2 n a n ) = 2V = 4a 2 . 
If n — 3, it becomes 

(2 V) = 2 3 a 3 = So? = 2aX2aX 2a. 

If 2a be negative by actual multiplication, then 

— 2a X —2a — 4a 2 , 
but — 2a X — 2a X — 2a = — 8a 3 , 
and — 2a X — 2a X — 2a X — 2a = 16a*, and so on. 

That is, inductively, All powers of a positive monomial 
are positive ; and, All even powers of a negative monomial 
are positive, and all odd powers are negative. 

Questions to be Answered or Found Out by the Learner. 

(a 2 ) 3 = what ? (a 3 ) 2 = what ? (a m ) n = what ? 

If m = 4, n = 3, a = 2, find the value of (a m ) n . 

Is (a w ) n = a m + * ? Why ? If not, what should it be ? 

What is the difference between 3a m/n to the mth. power, 
and 3a w/n to the nth power? 

Polynomials. 

It has previously been shown how to square (a + b) and 
(a — b). This process may be extended so as to include 
any number of terms without performing the actual work. 

Thus, 

(a + b) 2 = a 2 + 2ab + l\ 
and 

(a - If = a 2 - 2ab + b\ 



146 ALGEBRA. 

Again, 

(a + b + c) 2 = a 2 + b 2 + c 2 + 2aS + 2«c + 2fo, 
and 

( a -f J 4- c + rf)« = ^ + I 2 + c 2 + d 2 + 2a& + 2ac + 2ad 

+ 2be + 2bd + 2cd. 

An examination of the results will indicate the law ob- 
served in writing the complete product; hence to square a 
polynomial, — Add to the square of each term twice the 
product of that term and every term that follows it. 

To find (a + b — c) 2 , we simply change a 2 + b 2 -\- c 2 -f- 2ab 
+ 2ac + 2bc to a 2 + b 2 + c 2 + 2ab - 2ac - 2bc. 

By changing the sign of any letter a new formula is 
obtained; but it must be remembered that the even power 
of a negative quantity is positive. Again, in these expan- 
sions the teacher should not fail to direct the learner's 
attention to the ease with which polynomials can be raised 
to higher powers without recourse to actual multiplication. 
Laws stand far above all empirical processes. 

(a + by = a + b; 

(a + b) 2 = a 2 + 2ab + b 2 ; 

(a + bf = a* + 3a 2 b + 3ab 2 + b 3 ; 

(a + by = a* + ±a s b + 6a 2 b 2 + ±ab 3 + S*; 

(a + b) 5 = a 5 + 5a"b + 10a s b 2 + 10a 2 b 2 + bab* + b\ 

The learner will observe the following particulars : 

1. The leading letter decreases its power regularly from 
the left toward the right, while the other increases regu- 
larly. 

2. The sum of the exponents in each term is constant. 

3. The coefficient of the first term is one, and the co- 
efficient of the second term is the exponent of the bino- 
mial. 

4. The coefficient of any term is found by multiplying 
the coefficient of the preceding term by the exponent of the 



POLYNOMIALS. 147 

leading letter of that term, and then dividing the product 
by the number of the term counting from the left. 

5. If the binomial be of the form {a — b), then in the 
expansion the odd terms will be positive, and the even terms 
negative, counting from the left. 

Again, if we observe the coefficients in the expansions, 
we have the following: 
(1) a + b =1 + 1 = 2; 
(8) (a + bf=l + 2+l = ±; 

(3) (<*+'*)»= 1 + 3 + 3 + 1 = 8; 

(4) (a + bf = 1 + 4 + 6 + 4 + 1 = 16; 

(5) (a + b) b = 1 + 5 + 10 + 10 + 5 + 1= 32* 

(6) (a + b) Q = 1 + 6 + 15 + 20 + 15 + 6 + 1 = 64; 

(7) {a + b) 7 = 1 + 7 + 21 + 35 + 35 + 21 + 7 + 1 = 128. 
Hence, the sum of the coefficients forms a geometrical 

progression whose constant ratio is 2 ; but if either a or b 
be negative, then the sum of the coefficients in each expan- 
sion is zero. 

Another property of the coefficients is still more remark- 
able than any yet mentioned. For instance, if we regard 
the first and last coefficients in each expansion to be one, 
then we can derive the other coefficients in the following 
simple manner, thus: 

For (1) = 1 + 1; 

and 

(2) =1 + 2 + 1; 

that is, we add the coefficients (1 + 1) of (1), which sum is 
the middle coefficient of (2). 

Again, adding (1 + 2) and (2 + 1), we have 

(3) = 1 + 3 + 3 + 1. 

Now, (1 + 3), (3 + 3), (3 + 1), and we have 

(4) = 1+4 + 6 + 4 + 1; 
for 

(5) = l + 5 + 10 + 10 + 5 + l, 
and so on, 



148 ALGEBRA. 

From this simple law the learner can write the expan- 
sion without any difficulty. 
Or it may be obtained thus: 

(i) = i + i; 

W ~ 1 + 1 = 1 + 2 + 1; 

/ov _ 1 + 2 + 1 

W ~ 1 + 2 + 1 = 1 + 3 + 3 + 1; 

m _ 1 + 3 + 3 + 1 

W~" 1 + 3 + 3 + 1 = 1 + 4+6 + 4+1; 

{ ks _ 1 + 4 + 6 + 4 + 1 

W- 1 + 4 + 6 + 4+1=1 + 5+10 + 10 + 5 + 1; 

, fi v 1 + 5 + 10 + 10 + 5+ 1 

W - 1+5+10+10+5+1 = 1+6+15+20+15+6+1. 

There is another modification of expansion when the 
polynomial consists of more than two terms which requires 
something more than a passing remark. The case is this: 
Suppose it be required to expand (a + b + cf. By actual 
multiplication it can be shown that 

( a + h + c f = a 3 + b s + c 3 + 3a 2 * + 3a 2 c + Wa + 35 2 c 

+ 3 &a + 3c 2 £ + 6a*c. 
Also, 
( a + J + c + tf) 3 = a 8 + & + c 3 + d s + 3a 2 * + 3a 2 c + 3« 2 J 

+ 3* 2 a + 3* 2 6-+3*W+3c 2 a + 3c 2 * + 36'W 

+ 3d 2 a + 3d 2 b + 3d 2 c + 6abc + 6aM 

+ 6aca* + 6bcd. 

The law is: Write the cube of each term, plus three times 
the square of each term into each of the other terms, and plus 
six times the product of all the terms, taken three at a time. 

A little practice and attention will enable any one to 
write without hesitancy the cube of any polynomial. The 
advantage of understanding how to employ "short methods 



RADICAL QUANTITIES. 149 

and to permute quantities in symmetrical expressions" will 
save both time and labor. Practice and theory are thus 
combined, and the theory is turned to practical account. 

With many the mental effort is not too great in expand- 
ing directly such expressions as 

{2a + 36) 3 , (3a 2 - 4S 2 ) 3 , (6a 2 x + Wyf, 
and so on. 

With advanced classes, or those of mature judgment, the 
teacher may very profitably call attention to the expansion 
of (a±b) n , and show how to apply it when n — 2, n = 3, 
n = 4, n = 5, and so on; or even when n is fractional. 

Such glimpses open the way to a better understanding of 
Newton's Formula than any mere abstract statement could 
possibly do. The very idea that the learner is going over 
the same line that Sir Isaac Newton once pursued will 
have an inspiriting effect. 

Radical Quantities. 

Exponents. 

The Theory of Exponents deserves more than a passing 
notice. Thus far the pupil has used integral positive ex- 
ponents. Since any quantity may be regarded as positive 
or negative, integral or fractional, so an exponent may be 
treated under any two of these four conditions. 

Things to Teach. 

1. Teach clearly that the numerator of the exponent 
indicates the power to which the quantity is to be raised, 
and the denominator denotes the root to be extracted. 

Thus, a^ — VaF denotes that a is to be "squared," and 
then the " cube root" extracted. 
Also, that 

eft = Va, a* = Va, eft = Va*, a™ = Vd*, 
are applications of this principle. 



150 ALGEBRA. 

2. The meaning of the following and similar exercises: 

eft x aft = cfi + * = a 1 = a; 
aft xai = a$+i= a 2 ; 

1 2 3 J- 4 7. h~r, 

a? x ft 3 = « 6 6 = a 6 = r a 7 ; 

4- 4 -1 6 4-_4_-|- 3 13 12/— :, 

a* X fl* X fl* = a" T i8 T " = a 1 * — yV 3 . 

n ^ — ™ _^. 2n 4- n 

3. ad = r a n ; ad X aw = a «<* 5 

?i w r wps 4- mds + rdp 

Cld X CIP X flf = a &* 

That is, the exponents are reduced to a common denomina- 
tor, and added. 

4. a 2 =- 2 ; a * = -^; a * = _. 

While many expressions having negative exponents fre- 
quently occur, yet the reciprocals of these expressions can 
he taken, and then the negative exponents become posi- 
tive, and are so treated. However, such quantities can be 
handled quite as easily as those having positive exponents. 

Quantities having negative exponents may be regarded 
as having their origin in a fractional expression; thus, a~% 
may be conceived as arising from 

2 



i 1 a 2 1 

«* = — ; or — = — = a"*; 



rfi a? a* aft 

or from any other form in which the denominator is aft 
greater than the numerator. 

Exponents convey a language of deep significance in 
mathematics, and are so important that one mistake in them 
will vitiate a whole discussion or demonstration. Mathe- 
matics prides itself on the brevity of its language and on the 
universality of its symbols. Yet every symbol must be 



RADICAL QUANTITIES. 151 

interpreted exactly. In one sense, exponents are instru- 
ments employed by algebraists to reduce heterogeneous, or 
refractory, quantities to homogeneous ones whenever it is 
possible. However, no new principle is employed in treat- 
ing exponents that has not been already used. Exponents 
are the " handspikes of Algebra," and by the aid of which 
dissimilar quantities may be grouped, separated, multiplied, 
or divided. 

Here, again, the learner's algebraic vocabulary must be 
enlarged by the introduction of a few new terms. There- 
fore it is necessary that he should learn the following for 
all time, namely: 

1. A Simple Radical Quantity: as, 3 Vs. 

2. The Radical Factor: V$. 

3. The Radical Coefficient: 3. 

4. The Degree of the Radical. 

5. When a Radical is in its Simplest Form. 

6. A Radical Quantity. 

7. An Irrational Quantity. 

8. Similar Radicals. 

9. Dissimilar Radicals. 

Reduction of Radicals. 

The Reduction of Radicals consists in changing their 
forms without altering values. For instance, it may be 
more convenient to change a complicated expression into a 
simpler one, or to render dissimilar expressions similar, or 
to change irrational quantities into rational ones. 

After knowing how to read Radicals, the simplest process 
is to put a quantity under the radical sign, or to take a 
quantity from under the radical sign. Simple exercises 
for purposes of illustration are always preferable, because 
less confusing to the pupil. 



152 ALGEBBA. 

Before proceeding further with this subject, the teacher 
should call attention to the method of reading exponents. 
Thus, " a d " is read, the dthpotver of a, or a to the dth power, 
if d is a positive integer; but if d be integral and negative, 
then it should be read, " reciprocal of the dthpotver of a/ 9 
or the reciprocal of " a " to the dth. power. If d be fractional, 
say, as "f," it should be read "a exponent d," "or a expo- 
nent §;" but never " two-thirds power " 

1. Familiarize the pupil with the process of putting co- 
efficients under the radical sign. A few exercises will be 
sufficient for this purpose. Then, let him take the same 
quantity out from under the sign. Select problems from 
the text. Practice makes perfect. Teach each step before 
advancing to the next. 

2. When the quantity under the radical sign is a common 
fraction, change it to an equivalent expression in which the 
quantity under the radical sign shall be entire. This is an 
invaluable transformation. 

3. Change radical quantities to their simplest forms, 
or reduce a radical quantity to a higher or lower degree. 
These reductions correspond to certain lower arithmetical 
operations which the pupil will readily perceive. Lastly, 
reduce radicals having unequal indices to equivalent quan- 
tities having the same indices. 

These three steps have for their object the preparation 
of radical quantities for addition, subtraction, multiplica- 
tion, and division. 

Eequire the learner to show upon what principle, defini- 
tion, or axiom each step depends in the solution of a 
problem. This is one of the very best mental disciplines. 
To follow out consecutively every principle, and trace it 
backward to its origin or dependence, cultivates a close 
habit of thought that will be invaluable in after life. It is 
to unroll the scroll 



Similarly, 



-RADICAL QUANTITIES. 153 

As an illustration of this suggestion let it be required — 
To find the sum of 

Solution. 

\ A*/ \ « 3 / \ « 3 / 

( a 2 + a*£* ) = a*( cfi -f 5* 1 . 

Whence, 

( c t + j*)(j* + «*)* = {(«* + 5 f ) 3 f*. 

The first step is to reduce the first quantity to an im- 
proper fraction. This depends upon what principle ? The 
second, to introduce a coefficient under the radical sign. 
Upon what does it depend ? What operation is involved ? 
How is it done ? Third, to suppress a common factor. 
W r hy ? What principle is involved ? 

Fourth, to take a factor from under the radical sign. 
How is it done ? Why is the value of the expression not 
changed ? 

Fifth, to multiply two factors dissimilarly involved and 
to indicate their product. 

Wlmt axioms are involved in all these operations ? Name 
them. How many of the original mathematical operations 
are involved in this solution ? 

In the solution of all Radical Quantities let thege truths 
be deeply impressed on the mind of the learner, — that dis- 
similar quantities must, if possible, be made similar ; that 



154 ALGEBRA. 

complicated expressions are to be put into simpler or sim- 
ple at forma ; that quantities, so far as possible, must be 
freed from indicated roots ; and that irrational quantities 
are to be rationalized. 

To enforce these truths when a problem is solved, the 
teacher will first call attention to each step as it is taken, 
and afterwards have the pupil or class explain. Skill in 
Algebra is secured by intelligent practice. If a boy is 
obliged to roll a heavy log which he cannot lift, he uses his 
inventive faculties to assist him. Handspikes and props 
are found, and he makes a practical use of the lever. So 
it should be in attacking a mathematical problem. The 
knowledge the learner already has of definitions, axioms, 
principles, and of operations previously employed are now 
laid under tribute to help him solve the questions proposed. 
In the case of the log, the boy knew exactly what he had 
to do; but what is intended in the enunciation of a problem 
may not be quite so evident to the reader, hence his first 
mental effort is to ascertain what is required; or to bring 
the proposed question under some form with which he is 
already familiar. 

There are two more points under Eadicals that require 
attention. I refer to Quadratic Surds and Imaginary 
Quantities. 

Under the first, aside from methods of solution, the fol- 
lowing principles lie at the foundation of the subject, 
namely: 

1. That no irrational quantity may be expressed by a 
fraction. Why? 

2. A quadratic surd cannot be equal to the sum of a 
rational quantity and a quadratic surd. Why ? 

3. If two quadratic surds cannot be made similar, their 
product is irrational. Why ? 

4. A quadratic surd cannot be equal to the sum of two 
dissimilar quadratic surds. Why ? 



RADICAL QUANTITIES. 155 

5. In an equation of the form x ± Vy = a ± Vb, % = ci, 
and Vy = V#. That is, the rational parts are equal, and 
also the irrational parts are equal. Kequired, the proof. 

In the treatment of "Imaginary Quantities/' the appar- 
ent deviation from ordinary rules of procedure appears in 
the case of Multiplication. Algebraists have resorted to 
various expedients in order to explain the apparent ambi- 
guity with more or less success. The questions that shake 
the learner's faith are these: If a quantity is imaginary, 
how can it become a real quantity ? Is there any way to 
tell how, for instance, the pupil shall know Va 2 = — a, 
unless he knew beforehand that a is minus before it was 
put under the radical. This brings him to a standstill 
when he remembers that the even root of a negative 
quantity is pronounced impossible. The seeming con- 
flict, Va 2 = + a, and Va 2 = — a, is reconciled when it is 
known that V— a X V — a = Va*. The trace of the im- 
aginary quantity is thus preserved. 

The safest and simplest direction to the pupil is to put each 
of the imaginary factors into the form of V — a 2 = a V — 1, 
and then apply the same rules as in other Eadical Quanti- 
ties. 

The following will illustrate this remark: 



1. Find the sum of V — 1 and 



Solution. V - 16 = 2 V - 1; 
and 



JTZ^ + j/__ 16 = t/_ 1 + 2 fr^j; = 3 ^j^ AnSt 



2. Prom 3 — V — 64 take 2 + f — 1. 

Solution. 3 — V — 64: = 3 — 8 V~T^; 
whence 3-8 ♦^~T-(2 + V=l) = 1-9 V~^l. Ans. 



156 ALGEBRA. 



3. Multiply 2 V^l by 3 V - 16. 

Solution. 2 \ r ^l = 2 ^1 f~T; 

3 f- 16 = 3 tT(3 ^T; 
whence 

2 f I V~^\ X 3 ^16" ^"^T = 6 V&i f^~I 

= 6 X 2( - 1) = - 12. Ans. 

4. Divide 14 - ^15 - (7 Vd + 2 ^5) V~l by 7 - ^^5. 
Solution. 14 - ^15 - (7 4 / 3 + 2 ^5) 4^1 |7- V^h 

14-2 1^5 4^1 9 < - ^"-3 

- 7 V^3 - VT5 

- 7 ^^"3 - Vl5. 

Radical Equations. 

In the solution of complex radical equations very much 
depends upon the judgment and skill of the operator. 
Solutions are frequently lengthened unnecessarily because 
the operator does not detect how to contract the work he 
has to do. No general direction can be given for the solu- 
tion of all problems under this head. If a problem pre- 
sents any difficulty, the first thing to be decided is, in what 
does the difficulty consist, and how can it be avoided ? To 
make this suggestion more forcible, I will solve a problem 
which presents some difficulty to the average pupil. 

Problem. 



V(l + a) 2 + (1 - a)x + V{\ - of + (1 + a)x = 2a. 

It is evident that the equation must be freed from radi- 
cals, whence, squaring both members, collecting terms, 
arranging, and reducing, we have 



V(l - a 2 ) 2 + 2a(l + 3a 2 ) + (1 - a 2 )x 2 =- (a 2 - 1) - x. 



QUADRATIC EQUATIONS, 157 

Squaring again, we have 

(1 - a 2 ) 2 + 2s(l + da 2 ) + (1 - a 2 )x 2 

= (a 2 - l) 2 - 2(a 2 - l)x + tf 2 . 

Omitting equal quantities, and dividing by x> we have 

fl¥ = 8a 2 £, and # = 8. 

The theory of the solution is this: To keep the radical 
quantities on the same side of the equality; then to throw 
off the radical sign; collect and arrange terms, until both 
members are free from radicals. 

The learner is at liberty to employ what he has previously 
learned in solving a problem unless he is restricted to a par- 
ticular form of solution. 

Quadratic Equations. 

The learner has now made sufficient progress in Algebra 
to take hold of equations in a more comprehensive manner 
than he has hitherto been prepared to do. 

If all quadratic equations appeared under the form of 
perfect squares in both members, then it would be unneces- 
sary to teach methods " of completing the square." How- 
ever, the philosophy underlying this subject should be 
clearly presented to the pupil or class, and the necessity 
that existed which caused analysts to search for such a prin- 
ciple, and without the discovery of which no great progress 
could have been made, ought to be clearly and distinctly 
emphasized. The typical quadratic equation, 

Ax 2 + Bx+ C=0, 

is not necessarily a perfect power, and unless it is it must 
be made one by "completing the square," which consists in 
adding the same quantity to each member of the equation 
in order to make the left-hand member a perfect power, 
while the right-hand member may be or may not be a per- 
fect power, 



158 ALGEBRA. 

With a class of beginners, it is far better to teach one 
method of completing the square before attempting to teach 
another. The learner must have within himself the means 
of testing his own work, and one method thoroughly taught 
and well understood gives him this test. 

Pursuing the same plan as elsewhere indicated, let the 
pupil begin with the simple exercises first. Then the first 
step in the solution is to put the equation into one of the 
four simplest quadratic forms. That is, the coefficient of 
x 2 is unity. Secondly, to add the square of half the co- 
efficient of x to both members. Thirdly, to extract the 
square root of both members. Fourthly, to solve the sim- 
ple equation. 

For a w T hile, at least, the pupils should actually complete 
the square; but further on they should omit that part of 
the work and write out the values of the unknown quantity 
from the original equation, except in very complicated 

Solution of the General Form. 

It is evident that the General Form, x 2 ± 2px = ± q, 
contains four, and only four, special forms. 
Taking the special forms in order, we have 

(1) x 2 + 2px = q; whence x = — p ± Vq + p 2 . 

(2) x 2 — 2px = q ; " x — p ± Vq + p 2 . 

(3) x 2 + 2px = - q; " x = — p ± V- q + p\ 

(4) x 2 — 2px = — q; " x = p ± V — q ~\- p 2 . 

After specialization follows generalization, and this logi- 
cal unfoldment is the chart placed in the teacher's hand by 
which he guides his classes successfully through Algebra. 

Since the general form of the quadratic embraces four 
special forms, let us examine the roots of these special 



QUADRATIC EQUATIONS. 159 

forms. In (1) and (3) the roots are the same except q 
under the radical; and (2) and (4) are the same with that 
exception. The difference between (1) and (2) is in the 
sign of p not under the radical, and a like difference exists 
between (3) and (4). In other words, the values are all 
alike except the algebraic signs of p and q. Consequently 
the question is narrowed to this: Can all these special forms 
be brought under one comprehensive statement which will 
include them all ? A trial statement will help to determine 
the matter. Here it is: For the value of the unknown 
quantity, write half the coefficient of the first power with 
its sign changed, plus or minus the square root of the sum 
of the second member and the square of half the coefficient 
of the first power of the unknown quantity. 

It will be seen that the symbolized values of the unknown 
quantity are very much more easily written out than ex- 
pressed in words. The value of the generalization is this, 
the pupil writes the value of the unknown quantity with- 
out taking the time and trouble to complete the square. 
He throws away his crutches and walks. 

As another illustration of the advantage arising from 
generalization, let us take the following: 

c:r ± 2px = ± q. 

Here again are four special cases: 

(1) c:r} + 2px = q ■ whence x — — ^ ~ r — . 

c 

(2) *? - 2px = q; « x = P±*F±£g. 

c 

(3) ca* + 2px = -q; " x = ~ p± V P* ~ £1 . 

c 

(4) ex* - 2px = - q; « x = P ± ^EE, , 

C 



160 ALGEBRA. 

From the above the pupil should write immediately the 
values of x, which may be expressed thus: The unknown 
quantity is equal to half the coefficient of x toith its sign 
changed, plus or minus the square root of half the coeffi- 
cient of x squared and the product of the absolute term by 
the coefficient of x, and then dividing the whole by the 
coefficient of x. 

Sometimes it is desirable to avoid the occurrence of frac- 
tions in Solving equations of the second degree. The Hindu 
Method will enable the operator to write out the values of 
x without going through the steps of multiplying, reducing, 
and evaluating. 

To solve a problem by this method, I select the follow- 
ing: 

Given ex 2 + ax = b to find x. 

Solution. Multiplying by 4c we have 4cV + kacx — ±bc. 
Adding a 2 to both numbers, we have 

4cC 2 x 2 + 4acx -\- a 2 — a 2 -\- 4bc. 
Extracting 

2cx + a = ± Va 2 + 4tbc] 



-a± Va 2 + 4bc 
whence x = . 

2c 

Let us take the general form ex 2 ± ax = ± b, and sepa- 
rate it into the four special forms; we then have 

(1) ex 2 + ax = b; whence x = 

(2) ex 2 — ax = b; " z = 

(3) ex 2 + ax= -b; " x = 

(4) ex 2 — ax = — b: " x = 





-a± Va? 


+ Uc 




" 2c 




a 


± Va 2 + Abe 




2c 






-a± Va* 


- 4tbc 




2c 




a 


± Va 2 - 


Uc 



THE ROOTS OF QUADRATIC EQUATIONS. 161 

Comparing these results with the equations from which 
they are derived, we are enabled to write the values of x at 
once from the equation as follows: 

The unknown quantity is equal to the coefficient of x with 
its sign changed, plus or minus the square root of the co- 
efficient of oc after it is added to four times the product of 
a? into the known term, and the whole divided by twice the 
coefficient of &. 

The Roots of Quadratic Equations. 

It took a long time in the development of the science of 
Algebra for men to discover any relations existing among 
the coefficients of the unknown quantity and the roots of 
that quantity. In an equation of one unknown quantity 
of whatever degree certain relations always exist, and a 
knowledge of these conditions helps us to understand the 
true nature of the equation. For instance, the equation 
x 2 + 2## = — b has two roots, 

x = — a ~{ Va? — b, and x = — a — Va % — b. 

1. The sum of thjese roots is — 2a, that is, the coefficient 
of x with its sign changed. 

2. (- a + Vrf^Ht) x(-a- Va? - b) = b, that is, the 
product of the roots is equal to the absolute term, or the 
zero power of x. 

The object of this discussion is intended to show the 
learner that every equation of the second degree containing 
one unknown quantity is composed of the product of two 
binomials, and that if we are able to discover these factors 
by inspection, it would be unnecessary to resolve the equa- 
tion by the roundabout method of completing the square. 

In all treatises on Algebra discussions on the nature and 
limits of the roots of the four special forms of Quadratic 
Equations will be found, and which are valuable points to 
be emphasized in teaching this part of the subject not only 



1(52 



ALGEBRA. 



for sharpening the learner's analytical powers in investi- 
gating conditions implied in the results, but also for the 
bearing on the Theory of Equations in general. The pupil, 
unless properly instructed, gets the idea that the coefficients 
and absolute term are accidental affairs in the mechanism 
of equations. He fails to see that we start with roots to 
make equations rather than that we have equations to find 
roots. Instead of taking the equation as the essence, it is 
the developed exponent of the root involved in it, and these 
are the objects the pupil is trying to find. To solve a prob- 
lem is the simplest kind of work compared to the discus- 
sion of the problem after it is solved. A problem is never 
thoroughly understood till its nature and limitations are 
fully comprehended. I will endeavor to put this in a still 
stronger light. Suppose any equation is reduced to one of 
the four special forms, then let the pupil or class answer 
such questions as: 

1. What in the sum of the roots? 

2. They are equal to what ? Illustrate. 

3. What is the product of the roots ? 

4. Equal what ? 

5. Are the roots real or imaginary ? 

6. Are the signs alike or unlike ? Why ? 

7. If unlike, what is the greater sign ? 

8. Under what conditions will the roots be real ? Imagi- 
nary ? When equal ? When numerically equal with oppo- 
site signs ? 

Special Artifices. 
Mathematicians have resorted to many artifices for the 
purpose of solving special forms of Quadratic Equations. 
These artifices consist in transforming or changing the 
equations into simpler forms which can be more readily re- 
duced than the original ones. It is in this special line of 
work that algebraic skill shows itself to the best advantage. 



THE ROOTS OF QUADRATIC EQUATIONS. 163 

Refractory equations bristle, it may be, with difficulties; 
and unless the operator is able to plan a successful scheme 
of reduction, he is baffled at every point. 

The following will indicate partly the meaning of tenta- 
tive processes resorted to by algebraists: 

Given x + y = 3, xy = p, to find the value of 



x 2 -f- y 2 , %* + 


y\ 


z* + y l , x 5 


+ f. 


in terms of s and p. 








The results are : 








x 2 -\- y 2 = 


s* ■ 


-2p; 




x? + y 3 = 


s s ■ 


- 3ps; 




x' + y* = 


s* • 


— ips* + %P 2 ', 




X> -\- y 5 = 


s>- 


- 5ps 3 + 5p 2 s. 





In the solution of such problems as the following these 
equivalents may be employed advantageously: 

x + y = 4| c x + y = ii 

a 4 + if = 82 \ ; \x s + y* = 407 

But such substitutions are not absolutely necessary, since 
these and other similar problems can be readily solved by 
combining the equations as they are given. 

When two quadratic equations are symmetrical with 
respect to the two unknown quantities, they may frequently 
be solved by substituting for the sum and difference of the 
two other quantities. 

Thus: 

x + y =a, 
x 5 -f- y 5 = b, to find x and y. 

Here put x = s + d, y = s — d; and s = — . 

Solution. 

x? = s 5 + bs^d + 10s 3 ^ 2 + 10s 2 d z + 5sd* + d\ 
y*=s 5 - bshl + 10s 3 d 2 - 10s 2 d z + 5sd* - d\ 



104 ALGEBRA. 

and .c 5 + if = 2s b + 20s*<P + lOsd* = b; 
or *. + *. = * + _ = - m ~. 

Completing the square and reducing, the value of d can 
be found, and then the values of x and y. 

But these equations are also easily solved in the follow- 
ing manner : 

(1) x + y =a; 

(2) x h + y b = b; 

b b 

(3) = (2) -r- (1) = z 4 - zty + z 2 ?/ 2 - ^ 3 + y* = - 

= a 4 + y* - xy{x 2 + # 2 ) + a 2 ?/ 2 = -; 



a±V ~bT~ 

W hence #?/ = . 

By combining with (1), the values of x and y are found. 
Given (1) 7? + y* = 13, and (2) a? + # 3 = 35, 
to find ^ and y. 
Put ^ + V — s > an( i ^y = J 9 * 

Solution. 

(3) = (1) = 6< 2 -2^=13; 

(4) = (2) = * 3 -3/;s = 35; 

(5) = (4) X 2 = 2.s 3 - 6/w = 70,; 



(4) 

(5) 

Hence 


= (1) 2 

= (4) 3 


= x 2 + y 2 = a 2 — 2xy; 

= x i + 2x 2 y 2 +tf = a 7 - ±a 2 xy + ±x?y 2 

= 3* + y* = « 4 — 4a 2 #?/ + 2.rfy 2 . 


(6): 


= (3) 


• 
= a* — 4a 2 xy + 2x 2 y 2 — xy(a 2 — 2xy) -\~ 

2 2 & 


(>)■ 


= (6) 


b h 
= 5# 2 ?/ 2 — 5a 2 #y = a*. 

6i> 



s 3 - 


-39s 


= -70; 








s 4 - 


-39s 2 


-70s; 








s 4 - 


- 14s 2 


= 25s 2 - 


- 70s; 






s 4 - 


- 14s 2 


+ 49 = 


25s 2 - 


•70s + 


49; 


s 2 - 


-7 = 


±(5s- 


% 






5,2, 


or - 


-7. 









TEE ROOTS OF QUADRATIC EQUATIONS. 165 

(6) = (3) X 3s = 3s 3 - 6ps = 39s; 
(7) -(6) -(5) = 

(8) = (7) X s = 

(9) = (8) + 25s 2 = 

(10) = (9) 4- 49 = 

(11) = vW) 

and s = 
x and y are found from x + y = 5 and xy = 6. 

Homogeneous equations of the fourth degree are easily- 
solved by making a simple substitution, as is illustrated in 
the following : 

Given (1) 3x 2 + xy = 68, (2) 4y 2 + %xy = 160, 
to find x and y. 

Put y.~ nx. 

Solution. 

(3) = (1) = 3x 2 + nx 2 = 68; 

(4) = (2) = 4nW + Znx 2 = 160; 

Clearing and reducing, n = f, and a: = ± 4, y = ± 5. 
The other values of x and y can be readily found. 

Very frequently the shortest cut in the solution of a 
problem is to look at it for several minutes for the purpose 
of seeing it well. 

The following is one of this character: 

Given (1) (a* + \)y = xy -f 126, 
and (2) (x 2 + l)y = x 2 y 2 - 744, to find x and y. 

(3) = (1) ~ (2) x 2 y 2 - 744 = xy + 126, 
and #y = 30, or — 29, 

A — 97 ± 1/6045 

and a; = 5, -J-, or 



y = 6, 150, or 



58 
1682 



97 q= ^6045 



166 ALGEBRA. 

I will now select a few miscellaneous problems to illustrate 
some of the methods adopted in solving complicated equa- 
tions. 

Find x, y, and z from the equations: 

(1) * + y + z = 27; 

(2) x 2 + y* + z 2 = 269; 

(3) X s 4- 'y % + z* = 2853. 
Solution. 

(4) = (1) z = 27 - a; - y. 

Putting this value in (2) and (3), we have 

(5) = (2) x 2 + xy + y 2 -27x-2'7y = -230; 

(6) = (3) x 2 y 4- xy 2 -27x 2 - 54a;?/ - 27y 2 + 729a; 4- 729y 

= 5610. 
Adding 27 times (5) to (6), 

(7) x 2 y + xy 2 - 27 xy = - 600; 

(8) = (5) y 2 = 27a; 4- 27y - xy - x 2 - 230. 
Substituting in (7), 

(9) a; 3 - 27a 2 + 230a; = 600. 

(10) = (9) x ix, and adding 9a; 2 - 2700a; + 2500 to each 
member, we have 

(11) (2a; 2 - 27a;) 2 + 100(2a: 2 - 27a;) •+ 2500 = 

9a; 2 - 300a; + 2500; 

(12) = VU 2x 2 - 21x + 50 = ± (3a; - 50). 
Taking the lower sign, x = 12. 

From (7), if - 15y = - 50, y = 10. 
From (1), 2 = 5. 

Find x and y from the equations 

(1) (a; 2 -a;z/ + 2 / 2 )(a^+a;y + y 2 ) = 336; 

(2) (a?— x*y-\- a?y 2 — xy 3 -]- y*)(a^-r- a% + $?y 2 + zy 3 + y*) — 

87296. 
Solution. 

(3) = (1) 3* + afy» + y* = 336; 

(4) = (2) X s + a^ 2 + aY + *Y + # 8 = 87296. 
Put a; 2 + y 2 = », a; 2 ?/ 2 = ^, 



THE BOOTS OF QUADRATIC EQUATIONS. 167 

and 

(5) = (3) 5*-^ = 336; 

(6) = (4) s 4 - 3ps 2 + f == 87296. 

Substituting the value of p from (5), and reducing, we 
have 

(7) = (6) s± - 336s 2 = 25600, 
and s = 20 = x 2 -\- y 2 . 

Putting this value in (5), 

p = 64 = a?^ 8 , or ^ = 8, 
whence x = 4, y = 2. 

Find a; from the equation 

(1) 3a? - 4a? + 6^ - 4a = 12. 

Multiplying the equation by x, it becomes 

(2) 3x 10 - 4a; 6 + 6a? - 4a? = 12a;; 

(3) = (2) 4a? + 12s + 4a? = 3x 10 + 6a^. 
Adding 6x? + 9 to each member, and 

(4) 4a? + 12a; + 9 + 4a; 6 + 6a? = 3a?° + 12a? + 9; 
or 

(5) = (4) (2a; + 3) 2 + 2a?(2a; + 3) = 3a; 10 + 12a? + 9. 
Adding a; 10 to each side, and we have 

(6) (2a? + 3) 2 + 2a?(2a; + 3)+ x 10 = 4a; 10 + 12a?+ 9; 
(?) = Vj6) (2a; + 3)+a?= ±(2a? + 3); 

whence x = V2. 

Find x and y from the equations 

(1) xy + (a? + if) (1 + xy + xhf + a?y + xtf) = 87 ; 

(2) xy(x i -\-y i )(x i -\-xy-\-y 2 )(x i -{- y 2 -f- xy-\- xy 3 -^- x?y) =1190. 

Solution. 

(3) = (1) (xy -{- x 2 + y 2 ) + a;y(a? + y*) + zy 

(a? + y 8 )(a;y+a g + y') = 87; 

(4) = (2) xy(x? + f) (xy + x* + ff + 

x*y*(x* + y*)*(ary + z* + */ 2 ) = 1190. 
Put v = zy + x? -4- y 2 , and w = xy(a? -f- «/ 8 ). 



168 ALGEBRA. 

Then 

(5) = (3) v + w + viv - 87, 

and 

(C) = (4) vw(v + 1v) = 1190. 

From (5) and (6), v = 7, w— 10. 

Also we have 

(8) x* + y* + xy = 7, 
and 

(9) xy(x 2 + y 8 ) = 10, 
whence x — 2, y == I. 

Find the value of a; in the equation 

(1) a;* - 2aar> - 2ato + b 3 = 0. 
Adding (# 2 + 2S)a; 2 to each member, we have 

(2) a 4 - 2aa; 3 + (a 2 + 2£)z 2 - 2ab + J 3 

= (a 2 -f 26)a; 2 . 

(3) = 4^(2), a? - aa; + b = ±a? 4V + 26, 
and a 2 - [a ± ( VaT+2b)x ] ' = - b; 
whence 

g = i{a± *V + b 2 )± V(2a 2 -2b±2a V(a 2 + 2b))\. 

The effect upon my mind when I first examined such 
problems was not very encouraging. I could not see how 
there could be any method running through the solutions, 
and as much as I could make out of the different opera- 
tions was a series of " extraordinary guesses." But, com- 
mencing work first with simple problems, I discovered 
method in the guesses, and I now proceed to unfold it by 
using easy illustrations. 

l. Given a 4 - 12^ + 47ar> - 72a; + 36 = 0, to find x. 
We can write this equation as follows: 

(2) a* - 12a: 3 + 36a; 2 + 11a; 2 - 72a; + 36 = 0; 
or 

(3) = (2), (a 2 - 6a;) 2 + 11a; 2 - 72a; + 36 = 0. 



THE BOOTS OF QUADRATIC EQUATIONS. 169 

It is evident that (3) is not yet a perfect square, but if 
we add x 2 to each member we have 

(4) {a? - 6x) 2 +-12 (a? - 6x) + 36 = x 2 . 

That is, the (x 2 — 6x) puts (4) into the regular quadratic 
form, and extracting the square root we have 

x 2 - 6x -f 6 = ±x; 
whence x = 1, 2, 3, or 6. 

2. Given x* — 2r 3 -\- x = 30, to find #. 
Writing it under the following form we have 

a* - 2,7? + a? - x? + x = 30; 
or 

(a* _ 3)2 _ (a? _ 3) — 30. 

Adding \ to each member and the square root taken, 
the values of x are 

Many biquadratic equations can be solved by adding a 
binomial square to each member. 

3. Given x* + 3a? + x? - 3x = 2, to find x. 
Adding x 2 to each member, we have 

(1) x* + Set? + x? + x 2 -3x = x 2 + 2; 
or 

(2) = (1), [a? + | *)"- g + to). = a» + 2; 

»=»-t('4)-('+i)=M^ 

Adding J to (3), we have 

(5) = V(4), * + **-*= ±(* + t)i 
whence x = 1, — 1, — 1, or — 2. 

4. Find a; from the equation 

(1) • 3? - lx a + 9a; 2 + -21x = 54. 



170 ALGEBRA. 

(2) = (1) X 4 + 25a; 2 , 4a;* - 28a; 3 + 49r> + 12a; 2 + 108a; 

= 25a; 2 + 216. 
Subtracting 150a; from each member, (2) becomes 

(3) (2a; 2 - Ixf + 6(2^ - 7*) 

= 25a* - 150a? + 216; 

(4) = (3) + 9, (2x 2 - Ixf + 6 (2a? - 7x) + 9 

= 25a; 2 - 150a; + 225; 

(5) = V(4), 2x 2 - Ix + ; 3 = ± (5a; - 15); 
whence x = 3, 3, 3, or — 2. 

5. Find a; from the cubic equation 

(1) x s - 8a; 2 + 19a; - 12 = 0; 

(2) = (1) xx, x* - Sx z + 19a; 2 - 12a; = 0; 

(3) = (2), (a; 2 - 4a;) 2 + 3(a; 2 - 4a;) = 0; 

(4) = (3) + h C* 2 ~ ^Y + H* 2 ~ 4s) + I = I ; 

(5) = 1/(4), x 2 - 4a; = - f ± f ; 
whence x = 4, a; = 3, a; = 1. 

6. Find x from the equation 

(1) a; 4 - 27a; 2 + 14a; + 120 = 0. 

(2) = (1), x± - 26a; 2 + 169 = a; 2 - 14a; + 49; 

(3) = V{2), a; 2 - 13= ±(a;-7); 
whence x = 3, 4, — 2, or — 5. 

From the six preceding solutions the learner will see that 
problems can be solved without much difficulty, and that 
proficiency is obtained by intelligent practice. Exercises 
may be selected from the problems in the various treatises 
on Algebra, and with a month or more of practice on such 
problems, after the completion of ordinary chapters on 
Quadratics, the learner will be able to solve many problems 
which would baffle him except by the ordinary theory of 
equations. Of course there are methods of solving numeri- 
cal equations of the third and fourth degrees; but the art 
of solving them by quadratic methods is one of the most 



BATIO AND PROPORTION. 171 

valuable disciplines in the entire scope of mathematical 
science. 

Let the exercises be simple at first, gradually growing 
more difficult — not so difficult as to cause discouragement, 
and both teacher and learner or class will soon be astonished 
at the progress made in a very short time. 

After completing any ordinary treatise on Algebra, I 
spent some time in having my classes solve such problems 
selected from various sources. From a knowledge of what 
average classes can do in this kind of work, I do not hesi- 
tate to recommend it unqualifiedly. 

Ratio and Proportion. 

A clear and precise notion of all definitions is the first 
thing to be mastered under this head. A definition should 
never be the least bit hazy in the mind. Its meaning should 
always be strong and pronounced. Make clean work then 
among the definitions. Every one should be thoroughly 
comprehended. Confusion of terms here denotes indis- 
tinctness of thought. 

The second is the demonstration of all propositions. Of 
course the learner when studying Arithmetic got a sort of 
running idea of Ratio and Proportion, and some of that 
knowledge he still has, and the danger now to be guarded 
against is his inclination to bring all propositions and prob- 
lems under one general proposition — the product of the 
extremes is equal to the product of the means — and then 
solve by algebraic equations. However, in the demonstra- 
tion of the propositions, the fixing process is best accom- 
plished by using numbers substituted for the quantities. 
Especially is this true of those who experience difficulty in 
imagining quantities to be numbers without making the 
actual substitution. The tendency of the mind undoubtedly 
is to reduce every new acquisition to a close connection 
with something previously known, and here the imaginary 



172 



ALGEBRA. 



process is still carried on, and in some cases, at least, with 
great difficulty. 

Another precaution is necessary, and it is — 
In proportion, look to it that all problems are solved by 
proportion. Unless this suggestion is strictly enforced, 
the learner or class may never see the beauty and necessity 
for doing so. When working in proportion, the problems 
must be solved according to the propositions of proportion. 
The most beautiful solutions we have in Algebra are those 
effected by an application of the principles involved in the 
theory of proportion. 

To illustrate my meaning, the following equations are 
selected: 



(1) Given 


xy = 24, 






(2) x*-if 


: (x - y) 3 


:: 19 : 


1, 


to find x and y. 


& + <cy + y k 


: x 1, — 2xy -f- \f 


:: 19 : 


1. 


What Prop.? 


Zxy 


x 2 -f- xy + y* 


:: 18 : 


19. 


What Prop.? 


xy 


x 2 -\- xy + y 2 


:: 6 : 


19. 


What Prop.? 


xy 


(x + yf 


.: 6 : 


25. 


What Props.? 


4xy : 


(x + yf 


:: 24: 


25. 


What Props.? 


(x + yf : 


(x — y) 2 


:: 25 : 


1. 


What Props.? 


x + y : 


x-y 


: 5 : 


1. 


What Prop.? 


2x : 


2y : 


: 6 : 


4. 


What Prop.? 


x : 


y 

2x 

y=r 


: 3 : 


2. 


What Prop.? 
What Prop.? 


Y 0), 


xy = 24; v 


whence x - 


= ±6; y=±4 



Teacher, require at each step the reason therefor, if you 
wish your classes to do the work properly. Proportion is 
so frequently used in other branches of mathematics that 
its importance can hardly be estimated from its limited 
treatment in Algebra. The groundwork, however, needs 
to be well laid in Algebra so far as our books teach it, and 
then continued and applied in the more advanced branches. 



SERIES. 



173 



Third. The Terms. 



Fourth. 
Fifth. 
Sixth. 
Seventh. 



1 



Series. 

The subject of series is usually passed over rather lightly 
by most teachers of mathematics, and more particularly by 
those who do not understand what an important part 
Series play in many questions occurring in other branches 
of investigation. As a preparation for entering fully upon 
the subject, the first thing is to get a good definition of 
"a series." 

Second. The Law of the Series. 

r a — First Term; 
I = Last Term; 
d = Common Difference; 
r — Ratio; 

n — Xumber of Terms; 
s — Sum of the Series. 
The Kind of the Series. 
Finite or Infinite. 
Converging or Diverging. 

Increasing or Decreasing by Differences or by 
Ratios. 
The most important special cases are those of arithmetical 
and geometrical series. With the five different terms hav- 
ing any three given, the other two can be found. That is, 
each of these two progressions will give twenty formulas. 
For arithmetical progression a valuable exercise for the 
learner is to derive all the other formulas from the equa- 
tions l== a + {n — l)d, and S 

As soon as he derives a formula he should substitute 
numerical values for the letters in the formula. In this 
manner each formula should be found, and some of the 
principal ones ought to be remembered, which does away 
with the too common practice of referring to the book for 
everything. 



(« + o | 



174 ALGEBRA, 

Geometrical Progression. 

This series, in an algebraic point of view, affords much 
greater scope for a display of skill than that of Arithmetical 
Progression. The formulas are more complicated, and 
four of them involve logarithms with which the learner is 
not yet supposed to be acquainted. However, the other 
sixteen formulas can be d'educed from the equations: 

I = ar n - 1 (1), and S = lr -^ y or %— — (2). 
v ' r — 1 1 — r 

In these equations a, r, n are known quantities. Again, 
the learner should work out the formulas and make the 
substitutions in the manner described under Arithmetical 
Progression. 

Another good exercise is to have the learner or class 
point out the corresponding or correlative formulas in these 
two progressions. 

Two problems are given to illustrate, in part, the nature 
of the algebraic processes required to solve such questions. 

1. The sum of five numbers in Arithmetical Progression 
is 25 ; their continued product is 945. Find the numbers. 

Solution. Let (x — 2y), (x — y), x, (a£+ y), x + 2y, be 
the numbers. Then 

(1) {x - 2y) + (x - y) + x + (x + y) + (x + 2y) = 25; 
or, x = 5. 

(2) (5 - 2y)(5 - y) 5(5 + y)(5 + 2y) = 945; 
or 4y* - 125?/ 2 = - 436; 

or y 2 = 4, y — ± 2. 

Hence the numbers are 1, 3, 5, 7, 9. Ans. 

2. The sum of six numbers in geometrical progression is 
1365, and the sum of the extremes is 1025. Find the 
numbers. 

Solution. Let the numbers be x } xy> xy 2 , xy z , xy* } xy\ 



GEOMETRICAL PROGRESSION. 175 

Then, by the nature of the series and the conditions of the 
problem, we have 

(1) ^!^- = 13 65; 

(2) x + xtf = 1025 = (if + l)x — 1025. 
Equating the values of x iu (1) and (3), we have 

1365(y - 1) _ 1025 _ 

if-1 -y° + V 

273 205 



(3) 

(4) = (3), 



f + tf + 1 tf-jf + tf-y + V 



or 






Seducing, y + - = --. 

Completing the reduction, the numbers are 1, 4, 16, 64, 
256, 1024. 

The following problem and solution are inserted because 
they lie beyond the usual list of problems in university and 
college algebras : 

3. The sum of seven numbers in geometrical progression 
is 127, and the sum of their squares is 5461. Find the 
numbers. 

Solution. Let x, xy, xy 2 , xy s , xy*, xy 5 , xy 6 be the num- 
bers. Then 

(1) ' x + xy + xy 2 + xy 3 + wf + x ^ + %lf = 127; 

(2) a? + tfy 2 + xy + xhf + xhf + xhf + x 2 y 12 = 5461. 

By formula, 

(3) - ^Z_? = 127; 



176 



ALGEBRA. 



(*) 



x 2 y* — x 2 _ 



y 



-i 



(5) 



-«** mfc$- 



5461: 



127' 



(6) = (5), or 



r 



y b + f-y 3 + f-y + 1 _ 43 



/ + / + .*/ + / + / + ^ + l 127 
id arranging, we have 



Clearing 

(7) 84/ - 170/ + 84/ - 170/ + 84/ - 170?/ + 84 = 0. 
This is a recurring equation. 

(8) = (7)-/, 84/-170/+842/-: 



Put 



K)« 



y y y 



Then substituting, 
- 85s 2 - 845 



(9) = (8), 42s 3 - 85s 2 - 84* = - 85; 

whence s = f, y = ^ a; = 1. 

The numbers are 1, 2, 4, 8, 16, 32, 64. 

The learner should familiarize himself with the various 
artifices employed by algebraists for representing series. 
No very definite rule can be laid down for the solution of 
all questions; but problems may be grouped, and the method 
of solution for that group ascertained, and so on for other 
groups. Here, however, as in all other cases, success de- 
pends upon the ingenuity of the operator whose skill 
enables him to do the work in the easiest and simplest 
manner. Geometrical progression affords a fine field for 
the algebraist to exercise all his powers. 

Harmonical Progression. 

This progression, which is usually lightly touched upon 
in elementary treatises, derives its importance primarily 
from its connection with musical sounds. That is, if 
a series of strings of the same substance, but whose 
lengths are proportional to the numbers 1, ■$■, ^, £, -J> 



SERIES. 177 

\, \, etc., and these strings be stretched with equal force 
or weights, and any two be sounded together, harmony, 
as it is called, is produced. 

Any three quantities as a, b, c, are in Harmonical Pro- 
gression when a : c :: a — b : b — c. 

From the nature of this series two definitions have been 
formulated as follows : 

1. Three quantities are in Harmonical Progression when 
the first is to the third as the difference of the first and 
second is to the difference of the second and third. 

2. Three quantities are in Harmonical Progression when 
their reciprocals are in Arithmetical Progression. 

The first definition is better expressed by the proportion 
itself than when converted into words, while the second 
definition is really a proposition susceptible of proof. 

Also Harmonic Eatio and Proportion branch off into 
modern Geometry. This extension, in connection with its 
relation to Music, is sufficient apology for reference to it 
in this place. 

Continuation of Series. 

There is no limit to the different kinds of series, and a 
very large treatise would not exhaust the subject. It fre- 
quently happens that in the solution of a problem approxi- 
mate results are all that can be obtained, and recourse must 
be. had to some known series to even approximate the value 
of the unknown quantity. The law of a series must always 
express two definite facts : 

1. The rate of the increase or decrease of the series ; 

2. The intervals of time at which its values are taken for 
the terms of the series. 

The learner should learn as much of the nature of series 
as is usually found in our best American text-books on this 
subject. 

After the general discussion following each special series, 



178 ALGEBRA. 

if the learner attempts to generalize the problems, he will 
find that they can be thrown into four principal groups: 

1. To find any required term of a series. 

2. To interpolate a term or terms of a series. 

3. To find the sum of a series. 

4. To revert a series. 
To get at the way a series is formed we may regard it as 

resulting from dividing the numerator of a fraction by its 
denominator, or from involution or evolution. Algebraic 
quantities are expanded into series in four ways : 

1. By the Method of Division. 

2. By the Method of Undetermined Coefficients. 

3. By the Method of Involution. 

4. By the Method of Evolution. 
The method of expansion by Undetermined Coefficients 

is one with which the learner ought to be very familiar. 
It also affords a good exercise in determining the coefficients 
in the series. In decomposing rational fractions it is also 
advantageously employed, and it is- often used for this pur- 
pose in the Calculus. 

The third and fourth methods are so closely connected 
with the Binomial Theorem that they naturally fall under 
it ; but the Multinomial Theorem for expanding poly- 
nomials is regarded as less complicated than the Binomial 
Theorem. 

Reversion of Series. 

Eeversion of Series is so frequently used in the solution 
of equations that all algebraists recognize its importance. 
The value of this process depends upon its application in 
the more advanced mathematics. Many elementary things 
must be learned in the lower branches ready for use in the 
higher ones ; that is, if the learner intends to study beyond 
the merest rudiments. 






TEE DIFFERENTIAL METHOD. 179 

The Differential Method. 

This method has for its object the summation of a series 
by ascertaining the successive differences of its terms. It 
is easily developed into a series. Then the close connec- 
tion between it and the Binomial Formula will be seen. 

The questions that claim most attention in the treatment 
of this series are : To find any term of a series by first de- 
ducing the formula ; to find the sum of any number of terms 
of a series by first deducing the formula ; the interpolation 
of terms in a series of equidistant terms ; and the applica- 
tion to piling balls. 

A few problems will now be solved illustrating several of 
the preceding series. 

l. Find the nth term of an harmonical progression. 

Solution. Let L be the last term; then we have 

L = I-m*- 1) (fLzJ) == «(»»-i)-g("-2) > 

a v '\ ab J ab 

Inverting, 



ab 



Ans. 



a(n - 1) - b{n- 2)' 

2. Expand 1 into a series. 
Solution. 

x 

1. z 7—; — = x + x + x + x + x, etc. 

1 — 1 + X ' ' J 

X 

2. ■ — — = x — x 2 + x z — x* etc. 

x + 1 — 1 

£ + 1 — 1 x xr x z 

3. Decompose 

13 + 21s + 2x 2 

I — ox 2 -\- iz* * 



180 ALGEBRA. 

Solution. Let 
13 + 21x + 2x* = A(l - x)(l - 4x 2 ) + B{1 + x)(l - ix 2 ) 
+ C(l - z 2 )(l - 2x) + Z)(l - x 2 )(l + 2s). 
Now assume 

X _L, X JL, X -fTy X "A - , 

and we have 

1 6 2 16 . 



1 + # 1 — x 1 -\- %x \ — 2x* 

4k. Find the sum of n terms of the series l(m + 1), 
2(ra + 2), 3(w + 3), 4(ra + 4), 5(m + 5), etc. 

Solution. By the order of differences, D f — (in + 3), 
D 2 = 2, D 3 = 0. 

o / i -i\ i ^(^ — 1) / , ox , w('7&— l)(ft — 2) 

... S = n(m + 1) + — g — *■ (m+ 3) + ~* — ^3 " 

^(w + 1)(1 + 2n + 3m) 



X2 



1X2X3 



Binomial Theorem. 

The learner already knows how to expand a binomial 
when the exponent is a positive integer. He should now 
be held to a rigorous demonstration of this beautiful 
theorem. 

The formula ought to be as well known to the pupil as 
the simplest theorem in the book. When he has commit- 
ted the formula after first demonstrating it, then he can 
make all necessary substitutions for the exponent whatever 
its character. In the same expansion, let n be assumed as 
integral and positive ; integral and negative ; fractional and 
positive ; fractional and negative. By whatever method 
the theorem is demonstrated, the fact should never be lost 
sight of, that the learner or class should always remember 



LOGARITHMS. 181 

the theorem itself. Some things are to be learned and 
remembered. This is one of those truths. Some sort of 
demonstration of it is found in all books. Newton, it is 
claimed, never demonstrated it. 

The one fact, {a + i) n = a n + na n " x b + - l ^~~ 1 ' a n -*& 

n(n-l){n-2) _ 33 n(n-l)(n-2)(n-3) _ 
+ 073 a b + 1.2.3.4 a b +t 

etc., should be at the finger's end, ready for use whenever 
needed. 

Logarithms. 

Great clearness is necessary in the outset in teaching- 
Logarithms. The definition is often not comprehended by 
a class of pupils, and no doubt many use a Table of Loga- 
rithms without having the remotest conception of what a 
logarithm is. In my opinion the obscurity comes from 
the lack of a sharp distinction between the measure of 
the terms of a quantity, and the measure of its factors. 
The measure of the terms of a number is the measure of its 
effect when added or subtracted, while the measure of its 
factors is found by multiplication or division, i.e., 12 X3 = 
36 ; or 12 -=- 4 = 3. Or a more obvious distinction is, that 
the measure of terms is effected by a coefficient, and the 
measure of the factors by an exyonetit. To contract com- 
putation in which numbers are used as factors, we have a 
Table of Logarithms. That is, the exponents are the loga- 
rithms. 

Those points requiring special attention are : 

1. The Definition of Logarithms. 

2. The Base of the System. 

3. Meaning of the equation a* = n. What is a ? oc? n? 

4. Why cannot 1 or — 1 he used as the Base of a System ? 

5. The difference between Characteristic and Mantissa. 

6. In any system the logarithm of the base is 1. Why ? 



182 



ALGEBRA. 



7. In any system whose base is greater than 1, the loga- 
rithm of is + <*• Why ? 

8. In a system whose base is positive, a negative quantity 
has no real logarithm. Why? 

9. The logarithm of a product is equal to the sum of the 
logarithms of its factors. Demonstrate. 

10. The logarithm of a quotient is equal to the remainder 
obtained by subtracting the logarithm of the divisor from 
that of the dividend. Demonstrate. 

11. The logarithm of any power of a number is equal to 
the product of the exponent of the power and the logarithm 
of the number. Demonstrate. 

12. The logarithm of any root of a number is equal to 
the quotient obtained by dividing the logarithm of the 
number by the index of the root. Demonstrate. 

13. How to find logarithms of numbers in a Table of 
Logarithms between given limits. 

14. What the characteristic must be ? When negative ? 

15. How to use such a Table? 

16. What is meant by " Arithmetical Complement"? 

17. How to change " our common system" to the Napier- 
ian system. The converse. 

18. Solution of numerical problems by Logarithms. 

19. The Solution of Exponential Equations. 

20. Application of Logarithms to the solution of Com- 
pound Interest and Annuity Problems. 

21. Discussion of Exponential and Logarithmic Series. 

22. To calculate Logarithms. 

Permutations and Combinations. 

Of late years a tremendous impulse has been given to 
these two subjects in connection with that of the Theory 
of Probabilities. 

As much as can be done in this connection is to direct 
the pupil how to acquire a knowledge of this important 



PERMUTATIONS AND COMBINATIONS. 183 

department of Modern Mathematics. The problems range 
all the way from Arithmetic through the Calculus. But 
in teaching the subject to beginners, it is preferable to be- 
gin with very simple arithmetical questions. By degrees 
lead up then to the algebraic problems. 

The teacher can illustrate the fundamental principles of 
Permutation and Combination by using either figures or 
letters. Teach Permutation only, till its nature is firmly 
fixed in the mind. Use two or three letters first, then 
four, and so on. Follow this by Combinations with the 
same objects. Then compare the two, and let the differ- 
ences be noted. As an illustration of the above suggestion, 
suppose it be required to find the number of permutations 
that can be formed of the letters a, b, c, taken three in a 
set. The total number of permutations is equal to the 
total number of possible arrangements. The sets are : abc, 
acb, bac, bca, cab, cba. The combinations without repeti- 
tion are: abc, acb, bac. This distinction is fundamental. 
'abc 
acb , 7 

7 Combinations: \ acb 

cab ( bac 

cba 

Having given a sufficient number of exercises' of this 
character, the first problems following will be included 
tinder this head: "To find the number of permutations of 

n things taken 1, 2, 3, 4, m in a set, " n > m" 

Suppose n equals 4. Then if we have a, b, c, d, and one 
letter at a time be taken, there are 

4 permutations — a-\-b-]-c-\-d = 4 

Two letters 4x3 " = . 12 

Three "4x3x2" = 24 

Four "4X3X2X1" = 24 

Total permutations, 64 



Permutations 



184 ALGEBRA. 

The permutations of the letters a, b are — ab, ba; the 
combination is ab. 

If a, b, c be permuted in sets of three, we have abc, acb, 
bac, bca, cab, cba; but the total number of combinations 
are abc, acb, bac. 

Permutation is placing a number of things in all orders 
possible, while combination is the number of different col- 
lections that may be made of a number of things, so that 
no two collections shall be the same. 

The number of permutations of which n letters are sus- 
ceptible is equal to the product of the natural numbers 
from 1 to n. Suppose n equals 3; then a, b, c, when per- 
muted, will be 

n(n - l)(w -2) =3X2X1 = 6. 

If we take three letters, as a, b, c, we can arrange them 
in sets of one, two, three, thus: 

Of one, a, b, c. 

Of two, ab, ac; ba, be; ca, cb. 

Of three, abc, acb; bac, bca; cab, cba. 

The number of permutations singly equals n. 

The number of permutations in sets of two equals 
n(n — 1). 

The number of permutations in sets of three equals 
n(n — l)(?i — 2). 

The law of extension is obvious. 

To find the law of combination, proceed thus: The num- 
ber of combinations of n things taken singly equals n. 

The number of permutations of n letters taken two at a 
time is n{n — 1). Since each combination admits of (1 x 2) 
permutations, there are (1 X 2) times as many permutations 

^i\Yl 1^ 

as combinations; that is, the combinations -~ —- . 

1X2 

If n letters or things be taken three at a time, the num- 
ber of combinations will be — ^ ^ — - — - , and so on. 

1 X /v X o 



PROBABILITIES. 185 

A great variety of interesting problems can be selected 
by the teacher and given to the class. Pupils are interested 
in all such questions because they can see a practical ap- 
plication of the principles underlying each operation. My 
own experience, too, is that pupils should master Permuta- 
tion and Combination quite thoroughly before commencing 
the Theory of Probabilities. The exercises in our more 
modern treatises on Algebra are sufficient at least to give a 
class a fair start, and no teacher of Algebra can afford to be 
ignorant of these subjects. They constitute a bright spot 
in the algebraic region. 

Probabilities. 

Under this heading definitions are important; thus: 

1. What is the probability that an event will happen? 

2. What is the improbability ? 

3. What is a simple probability? 

4. What is a compound probability? 

5. What is a favorable case? 

6. What is an unfavorable case? 

7. When is an event dependent? 

8. When is an event independent ? 

Principles. 

1. The probability that an event will happen is equal to 
the number of favorable chances divided by the whole num- 
ber of chances. 

Algebraically thus: — — , where a denotes the favora ble 

chances and b the unfavorable ones. If a equals b, then 

— — - = - as in the case of tossing up a coin. 
a + b 2 ? fe l 

2. That an event will not happen is denoted thus: 

b 
a+ b' 



186 ALGEBRA. 

3. That an event did or did not happen is denoted 

thus: 

b 
-\ j—7 == 1 — certainty. 



a -\- b a + b 

4. That of several events of which only one can happen, 
the chance that some one of them will happen is the sum 
of all the chances. 

5. That two independent events will both happen is the 
product of their chances of happening. 

I sincerely hope all teachers of algebra will make it a 
point to cultivate an acquaintance with the elementary 
principles of this beautiful science. It is indeed one of the 
most attractive fields now before mathematicians. 

There is much upon this subject in our newest and 
freshest text-books, and a great deal more scattered through 
our mathematical literature. It affords me great pleasure 
to refer to the many able and elegant solutions found in 
foreign and native periodicals upon this subject, contributed 
by the lamented Prof. E. B. Seitz, and by Dr. Artemus 
Martin, now of A\ r ashington City. These two distinguished 
mathematicians deserve great credit for their researches in 
this department. Dr. Martin has done, and is still doing, 
more to popularize sound mathematical scholarship than 
any other person on this continent. 

The only direction that I would give for teaching this 
subject is, to create an interest and then give the class prob- 
lems, beginning at first with very simple exercises. 

General Theory of Equations. 

So far methods of solving equations of the first, second, 
third, and fourth degrees have been discussed. But the 
reduction of equations of the third and fourth degrees is 
attended with no little difficulty. So complicated is the 
solution of a complete equation of the fourth degree, that 



COEFFICIENTS AND ROOTS. 



187 



algebraists try to avoid it when possible. Perhaps little is 
to be gained by the complete solution of complete equations 
of the fifth and other degrees, since methods of solving 
equations having numerical coefficients are well known. 

My object in this chapter is to help the learner find the 
real roots of numerical equations. 

The typical equation is 

x n + Ax"- 1 + Bx 11 -' 2 + Cx n ~ z + Dx n ~ k + . . . = 0. 

In this equation A, B, C, D, n are integral and all the 
exponents positive. 

Suppose that an equation needs to be changed to the typi- 
cal form, some or all of the following conditions may arise: 

1. To make the exponents positive. 

2. To make the exponents integral. 

3. To make the coefficient of x n unity. 

4. To make the other coefficients integral. 



Coefficients and Roots. 

Let a, b, c, d be the roots of an equation of the fourth 
degree, then we have (x — a)(x — b)(x — c)(x — d) = 0; 

that is, x — a = 0, x — b = 0, x — c — 0, x — d = 0. It 
becomes 



x A — a 


z? + ab 


- b 


' + ac 


— c 


+ ad 


-d 


+ be 




+ bd 




+ cd 



x 2 — abc 

— aid 

— acd 

— bed 



x -f abed 



= 0. 



The law is: 

Coefficient of x n equals 1. 

Coefficient of x n ~ x equals the sum of the roots taken yrith 
their signs changed. 



188 ALGEBBA, 

Coefficient of x 11 " 2 equals the sum of the products of the 
roots taken two at a time. 

Coefficient of x 11 - 3 equals the sum of the products of the 
roots taken three at a time with their signs changed. 

The Absolute term is the product of all the roots. If the 
degree of the equation is odd, the sign of the absolute term 
is changed. 

It will be seen that by an examination of the coefficients 
of an equation, frequently roots may be found, and then 
the degree of the equation depressed. 

An excellent. exercise is to give the roots of an equation 
to a class to form the equation. 

First, let a = 3, i == 4, c — 5, d — 6. Secondly, 
a = — 3, i = 4, c = — 5, d = 6. Thirdly, take three 
of the roots negatively. Fourthly, all the roots negatively. 

In each case, let the pupil or class note the differences in 
the signs of the coefficients. 

Give exercises till these relations are well known and can 
be readily and quickly applied. 

1. If any term is wanting, its coefficient is 0. 

2. If the second term is wanting, the sum of the roots is 0. 

3. If the third term is wanting, the product of the roots, 
taken two at a time is 0. 

4. If the last term is wanting, one of the roots is 0. 

5. The last term is divisible by each of the roots. 

Exercises. 

Given x 3 — 3x 2 — 10^ + 24 = 0. Find the three roots 
by factoring the absolute term, 24. How many roots are 
minus ? Why ? 

In the equation x^ — 12x? + 4&c 2 — 68x + 15 = 0, two of 
the roots are 3 and 5; how may the other roots be found ? 
How is the depression of this equation effected ? 



OTHER SIGNIFICANT PROPERTIES. 189 

Other Significant Properties. 

1. Surd roots of the form a ± t, or imaginary roots of 
the form ± b V — 1, enter equations by pairs; hence every 
equation of an odd degree must have at least one real root. 
Why ? 

2. In the equation a? - &a? - 7a? + 29./' -f 30 = 0, how 
many positive roots are there ? Why ? How many nega- 
tive roots? Why? What is the law for the number of 
positive roots in an equation ? How can it be told by look- 
ing at an equation ? If any term or terms be wanting, how 
tell them ? Can the number of positive roots exceed the 
number of variations of signs ? What must be the sum of 
variations and permanences in any equation ? 

3. If all the terms of an equation be positive, there is no 
positive root. Why ? 

Has the equation a? - 11a? + ±±<: 2 — 76;/; + 48 = any 
negative roots ? Why ? 

If all the terms of an equation be negative, how many 
positive roots are there ? Why ? 

How many positive roots has the equation x 1 — 1 = 0? 
What is the test ? What signs will be given to the wanting 
terms in the equation ? If two or more successive terms of 
an equation be wanting, what inference in regard to imagi- 
nary roots ? 

•4. To determine whether an equation has equal roots. 

This is ascertained by rinding the greatest common divisor 
of the equation and its first derived polynomial, or differ- 
ential equation. If there is no common divisor, the equa- 
tion has no equal roots. Suppose that in testing for equal 
roots a divisor of the form (x — Vf(x + 3) 5 i g found; then 
(x — l) 3 indicates three roots equal to 1, and five roots 
equal to — 3. 

5. Superior and inferior limits of the roots of an equation. 

Determining these limits may not assist a great deal in 



15)0 ALGEBRA. 

finding the roots of an equation, yet they are guides beyond 
which the operator may not look. To know that the 
greatest negative coefficient increased by unity is greater 
than the greatest root of the equation, is some knowledge; 
if not very definite, yet it is worth knowing. 
Let us take the equation 

a 5 + 5a 4 + 2a 3 - 14a 2 - 26a + 10 = 
to find the superior limit of the superior roots. By formula 



i + 



n-m i 



n equals degree of the equation, m equals the degree of x 
in the highest negative term, and P the greatest negative 
coefficient. In this problem 

n = 5, ' m = 2, P = 26 ; 

.-. L = l+ ^7 26 = 1 + ^26. 

To find the limits of the negative roots, substitute — x 
for x, and proceed as in the case of finding the positive 
roots. 

Let the teacher select a group of equations such as the 
following: 

1. a 4 - 10a 3 + 35a 2 - 50a + 24 = 0, 

2. x? - 13a 2 + 56a - 80 = 0, 

3. a 4 - 6a 3 + 5a 2 + 12a = 0, 

4. 12a 4 + 55a 3 - 68a 2 - 185a + 150 = 0, 
and question the class on each equation, thus : 

How many roots has equation (1) ? What is their prod- 
uct ? What is their sum? What numbers may be its 
roots? Why? What are the limits of its roots? Has it 
an even or odd number of real roots? How many of its 
roots are positive? How many are negative? Has it equal 
roots? Why? Has it imaginary roots? Changing the 



STURM'S THEOREM— HORNER'S METHOD. 191 

sign of every other term in the equation, what is the effect 
on the roots? If the signs of all the roots of an equation 
be changed, how is the equation affected ? 

Sturm's Theorem. . 
This theorem is employed when it is desirable to ascer- 
tain the number and situation of all the real roots of an 
equation. The equation should be freed from equal roots, 
but if it be not freed from the equal roots, it will still give 
the number of distinct roots without repetitions between 
the assigned limits. Sturm's Theorem is used chiefly in 
finding the situation of the incommensurable roots of nu- 
merical equations. 

Some Properties of this Theorem. 

1. No two consecutive functions can become zero for the 
same value of x. 

2. When any function after the first vanishes, the two 
adjacent ones have opposite signs. 

3. If, as x increases, f(x) passes through zero, Sturm's 
functions lose one change of sign. 

4. If any of the other functions vanish, there is neither 
loss nor gain in the number of changes of signs. 

5. The total number of roots of J\x) will be found by 
substituting -|- co and — oo in the first term of each of the 
functions. 

6. If the first terms in all the functions after the f(x) 
are positive, then all the roots are real. 

7. If the first terms are not positive, then for every change 
of sign there are two imaginary roots. 

Horner's Method. 

I will introduce this method by quoting an extract from 
Prof. Augustus De Morgan, who did so much to popularize 
Horner's discovery in England : 

"Another instance of computation carried paradoxical 



192 ALGEBRA. 

length, in order to illustrate a method, is the solution of 
x z — 2x = 5, the example given of Newton's method, on 
wihch all improvements have been tested. 

"In 1831, Fourier's posthumous work on equations 
showed 33 figures of solution, got with enormous labor. 
Thinking this a good opportunity to illustrate the superior 
method of W. G. Horner, not then known in France, and 
not much known in England, I proposed to one of my 
classes, in 1841, to beat Fourier on this point, as a Christmas 
exercise. I received several answers agreeing with each 
other, to 50 places of decimals. In 1848 I repeated the 
proposal, requesting that 50 places might be exceeded : I 
obtained answers of 75, 65, 63, 58, 57, and 52 places. But 
one answer, by Mr. W. Harris Johnston, of Dundalk, and 
of the Excise Office, went to 101 decimal places." 

Again I quote from De Morgan : 

"It was somewhat more than twenty years after I had 
heard a Cambridge tutor show some sense of the true place 
of Horner's Method, that a pupil of mine who had passed 
on to Cambridge was desired by his college tutor to solve a 
certain cubic equation — one of an integer root of two figures. 
In a minute the work and answer were presented, by Hor- 
ner's Method. ' How ! ' said the tutor ; ' this can't be, you 
know.' ' There is the answer, sir,' said my pupil, greatly 
amused, for my pupils learnt not only Horner's Method, 
but the estimation in which it was held at Cambridge. 
' Yes,' said the tutor, c there is the answer, certainly; but 
it stands to reason that a cubic cannot be solved in this 
space.' He then sat down, went through a process about 
ten times as long, and then said with triumph : ' There ! 
that is the way to solve a cubic equation.' " 

The discovery of this method and its application to the 
solution of numerical equations of the higher degrees, as a 
labor-saving device, is excelled only by Logarithms. In the 
solution of numerical equations, unless I oa,n reduce them 



LOCI OF EQUATIONS. 193 

by some tentative process or by factoring, I use Horner's 
Method invariably. It can be taught to a class in Arith- 
metic just after 'Cube Boot' is learned." 

To teach it in Algebra, give a cubic wanting the second 
and the first power of the unknown quantity. Then when 
this class of problems can be easily solved, introduce a com- 
plete cubic of this form, x 3 -f- x 2 + # = 6,000 ; to be fol- 
lowed by cubics of still greater difficulty. 

The same plan should be pursued in the solution of 
equations of the fourth degree. About the only difficulty 
the learner experiences is in placing the numbers in their 
proper places in the columns each time. 

Both in cubics and biquadratics, the learner should find 
the negative roots as well as the positive ones. 

Proceeding in a precisely similar manner, take up higher 
equations. It is not till the pupil understands how to find 
the number and the situation of the roots of any numerical 
equation, and then is able to solve it, that he is prepared to 
handle algebraic problems. It is a fitting climax to ele- 
mentary Algebra for the learner or class to be thoroughly 
grounded in the most important principles involved in the 
philosophy of equations. 

Loci of Equations. 

An interesting phase of Algebra is to represent equations 
by diagrams or figures. While the subject belongs properly 
to another branch of mathematics, its introduction has a 
tendency to lead the learners onward to a higher conception 
of the equation. As soon as they discover that an equation 
of the first degree is the equation of a straight line, and 
that an equation of the second degree represents a curve, 
they naturally look upon an algebraic equation as an ex- 
pression whose properties can be revealed, constructed, and 
interpreted. 



194 



ALGEBRA. 



If the teacher desires his class to construct the loci of 
some equations, the following terms need to be learned : 

1. The Axes of Keference. 

2. Their Origin. 

3. The Axis of Abscissas. 

4. The Axis of Ordinates. 

5. The Abscissa of a Point. 

6. The Ordinate of a Point. 

7. The Fort of an Ordinate. 

8. The Coordinates of a Point. 

9. The Locus of an Equation. 

10. Constructing its Locus. 

11. Abscissas — positive and negative. 

12. Ordinates— positive and negative. 

13. The names of the parts into which the plane is 
divided by the axes. 

14. Counting Directions. 

The first step is to teach the learner to draw the axes, 
and then how to locate points in the four quadrants. 
2d. To construct such equations as 

y = 2x + 2, x = 2y — 3, y = mx -f< b. 



3d, Ax — by = 5, 

6x + 12y = 78. 

4th. x + y = 9 and 

x + y = 7 and 
x — y — 10 and 

5th. x 2 + 3x - 10 = 0, 
& - 20z 2 



3x + 2y = 21, 
2x + 12y = 40. 
x 2 + y 2 = 53, ' 
a? + 2/= 34, 
z 2 + ^ = 178. 
23 _ 2x 2 + 1 = 0, 
+ 64 = 0. 



GrEOMETEY. 



Historical Sketch. 

The origin of geometry is hidden in the past. The 
derivation of the word, from the two Greek words, " ge" 
and " metron," signifies " earth-measuring," or, more popu- 
larly, "land-measuring." Herodotus says, in speaking of 
Sesostris, King of Egypt: "They said also that this king 
divided the country amongst all the Egyptians, giving an 
equal square allotment to each, and from this he drew his 
revenues, having required them to pay a fixed tax every 
year; but if the river happened to take away a part of any 
one's allotment, he was to come to him and make known 
what had happened; whereupon the king sent persons to 
inspect and measure how much the land had diminished, 
that in future he might pay a proportionate part of the 
appointed tax. Hence land-measuring appears to me to 
have had its beginning, and to have passed over into Greece; 
for the pole and the sun-dial, and the division of the day 
into twelve parts, the Greeks learnt from the Babylonians." 

Diodorus re-enforces this statement as follows : " The 
river, changing the appearance of the country very materi- 
ally every year, causes various and many discussions among 
neighboring proprietors about the extent of their property; 
and it would be difficult for any person to decide upon 
their claims without geometrical proof." 

Eouche and De Comberousse, in the preface to "Traite 
de Geometrie Elementaire," maintain that the ideas of 
extent, position, and form are as ancient as the race, and 

195 



196 GEOMETRY. 

they attribute to the Egyptians and the Chaldeans the first 
attempt to co-ordinate these ideas. 

Thales of Miletus, in Asia Minor, born about 640 B.C., 
introduced geometry into Greece from Egypt, where he had 
been instructed by the priests. Upon his return to Greece 
he founded the Ionian School of Philosophy. He pre- 
dicted the eclipse of the sun which occurred during a 
battle between the Medes and Lydians about the year 609 
B.C. He discovered that all angles in a semicircle are 
right angles, and he demonstrated some propositions re- 
lating to the similarity of triangles. The height of the 
pyramids he measured from their shadows, and by an ap- 
plication of the principles of geometry he could tell the 
distance of vessels remote from the shore. Many of his 
propositions were afterward collected in Euclid's Elements. 

Pythagoras of Samos, a disciple of Thales (580 B.C.), 
founded a celebrated school in Italy that bore his name. 
He had studied in Egypt, spent some time in Babylon, and 
perhaps visited India prior to his residence in Italy. To 
him are attributed the discovery of the incommensurabil- 
ity of the diagonal and the side of a square, that the square 
on the hypothenuse of a right-angled triangle is equivalent 
to the sum of the squares on the other two sides, that the 
circle has the maximum area of any plane figure having 
the same perimeter, and the sphere the maximum volume 
bounded by a given surface. He was the first to investi- 
gate the properties of regular polyhedrons, and one of his 
pupils solved the problem of finding two mean proportion- 
als between two given straight lines. 

Hippocrates, of the island of Chios, who lived about 400 
B.C., was one of the most noted Greek geometers of an- 
tiquity. He was the first to effect the quadrature of a 
curvilinear space by finding a rectilinear space equivalent 
to it; and he demonstrated that the crescent bounded by 
half the circumference of one circle and one fourth the 






HISTORICAL SKETCH. 197 

circumference of another is equal to an isosceles right- 
angled triangle whose hypothenuse is the common chord 
of the two areas; also, that the duplication depends upon 
finding two mean proportionals between two given lines. 

Plato, the philosopher, was one of the most distinguished 
promoters of the science. He introduced the analytical 
method of investigation, and discussed some properties of 
conic sections and of geometrical loci. In the school 
which he established, the duplication of the cube and the 
trisection of an angle were first investigated. The first of 
these two noted problems Plato solved; but the second has 
thus far baffled all attempts at a solution by elementary 
geometry. Eudoxus, who lived at the same time with 
Plato, found the volume of the pyramid and cone, and 
made considerable proficiency in conic sections. 

Plato's disciples gave a great impulse to the science of 
geometry. Euclid, who belonged to the famous school of 
Alexandria, had studied at Athens under the followers of 
Plato. This remarkable school was established about 300 
B.C. Perhaps fifteen years later, Euclid collected and 
systematized all the truths, propositions, and theorems 
then known as elementary geometry, and to which he added 
many new ones. This treatise is known as "Euclid's 
Elements." In reply to King Ptolemy, who had asked 
if there was no easier way to learn geometry than in 
his "Elements," Euclid promptly said: "No, sir ; there 
is no royal road to geometry." The method of proof 
known as the rednctio ad absurdum first appeared in 
"Euclid's Elements." Many of Euclid's writings were 
lost, the most important one of which was his treatise on 
prisms. The " Elements" is the work that is best known. 
It is composed of thirteen books, which treat of geometry 
and arithmetic. This work has been translated into the 
languages of all nations that have made much progress in 
the sciences and arts, and has been more generally used in 



198 



GEOMETRY. 



teaching than any other text-book ever written. Of the 
thirteen books composing "Euclid's Elements," the first 
four and the sixth treat of figures in a plane, while the 
fifth treats of proportion; the next four belong to arith- 
metic, and incommensurable quantities. The eleventh and 
twelfth treat of solid geometry, and the thirteenth book 
relates to the five regular solids. Two additional books, 
once credited to Euclid, were doubtless added about two 
hundred years later. School editions usually embrace the 
first six and the eleventh and twelfth books. 

Archimedes, born at Syracuse 287 B.C., one of the most 
distinguished geometers of antiquity, wrote two books on 
the sphere and cylinder. He demonstrated that the 
sphere is two thirds of the circumscribing cylinder, whether 
their surfaces or volumes be compared. He determined 
the circumference of a circle to be between 3-fg- and 3-^-f 
times greater than its diameter; he compared the area of 
the ellipse with that of the circle, and proved that the area 
of any segment of a parabola cut off by a chord is equiva- 
lent to two thirds of the circumscribing parallelogram. 
His line of work was in the direction of metrical geometry. 

Apollonius was a profound and original geometer, born 
at Perga in Pamphylia about 250 B.C. His writings re- 
late to the geometry of form, the conic sections in which 
he developed the properties of asymptotes, foci, conjugate 
diameters, normals, the theory of polars, and the primary 
notions of maxima and minima; and the celebrated theory 
of cycles and epicycles, so long employed in explaining the 
apparent movements of the heavenly bodies, is attributed 
to him. His principal work, the treatise on conic sections, 
contained eight books, seven of which are still in existence. 
This work was written in Greek during his residence at 
Alexandria. An Arabic version of one of his treatises is 
preserved. The title of "Great Geometer" was given 
him. 



HISTORICAL SKETCH. 199 

The successors of Archimedes and Apollouius directed 
their studies to those sciences which had a particular hear- 
ing on the science of astronomy. Also, about 150 B.C. 
flourished Hipparchus, the great astronomer of antiquity, 
who is regarded as having discovered the method of pro- 
jecting the sphere stereographically, and of having invesi- 
gated the properties of transversals in both plane and 
spherical triangles. 

From this date till the discoveries of Pappus, who lived 
at Alexandria about 380 or 400 a.d., geometrical investi- 
gation had virtually ceased. Pappus announced the prin- 
ciple of the famous rule now known as " Guldin's Theo- 
rem;" he gave the first example of the quadrature of a 
curved surface; the fundamental principles of the anhar- 
monic relation; the germ of the theory of involution; and 
the property of the hexagon inscribed in a conic section. 

Hypatia, the celebrated daughter of Theon, displayed 
even greater talents than her father, whom she succeeded as 
teacher of mathematics at Alexandria near the close of the 
fourth century. She wrote commentaries on Apollonius 
and Diophantus. Her works were destroyed when the 
Mohammedans burned the library of Alexandria. The 
Alexandrine school ceased when the city was conquered 
by the Arabs. With the fall of Alexandria in 638 A.D., 
another school sprang up at Bagdad. A few able com- 
mentators had access to some writings of the Greeks 
which had escaped the disastrous conflagration at Alexan- 
dria; but in Europe a profound stagnation prevailed for a 
thousand years, which clearly divides the ancient geometry 
from the modern. After the revival of learning "Euclid's 
Elements " were first made known in Europe through the 
medium of an Arabic translation. 

Vieta, the veritable creator of algebra, applied this 
science to the solution of problems in geometry. He con- 
structed graphically the roots of equations of the second 
and third degrees, and was the first to solve the problem 



200 GEOMETRY. 

of drawing a circle tangent to three given circles; but the 
modern methods of solution are more elegant and simple. 
We are indebted to Vieta for the new and fruitful idea of 
the transformation of spherical triangles; and his reciprocal 
triangle, without doubt, conducted Snellius to the discov- 
ery of the supplementary triangle. The writings of Kepler 
(1571-1631) and of Fermat (1570-1633) contain the germs 
of the method of infinitesimals. We owe to Kepler, the 
founder of modern astronomy, the treatment of the circle 
as composed of an infinite number of triangles, having 
their vertices at the center, and the cone as composed of 
an infinite number of pyramids, all having the same vertex 
as the cone; and to Fermat, the restriction of the plane 
surfaces of Apollonius, and the first complete solution of 
the problem relating to the contact of spheres. 

Pascal, so well known on account of his works on the 
cycloid, indivisibilities, and the calculus of probabilities, 
discovered, at the age of sixteen, the beautiful proper- 
ties of the "mystic hexagramme," or Pascals Theorem, 
which he took for the basis of his complete treatise on 
conies. A skeleton or outline of his works, as an essay 
on the conies, was published in 1640. From the writings 
of Pascal is recognized the influence exerted upon his con- 
temporary Lyonnais Desargues (1593-1662), who was one 
of the most skillful geometers of that age, and whom M. 
Poncelet called the Monge of the century. The ancient 
geometricians studied the conic sections from the cone it- 
self, and employed tedious solutions for each of the curves. 
Desargues referred directly all properties of the conies to 
the circle at the base of the cone, regardless of what the 
forms might be. Among the discoveries he made were the 
inscribed quadrilateral in a conic, the fundamental prop- 
erties of homologous triangles, and an extension of the 
properties of the circle so as to include all classes. His 
demonstrations were broad generalizations. 



HISTOBICAL SKETCH. 201 

Descartes was born in 1586 and died in 1650; he pro- 
duced a complete revolution in geometrical methods by 
bringing geometry under the domain of algebraic treat- 
ment, thus founding that branch of mathematical science 
called Analytical Geometry. Owing to the universality of 
his solutions and the comprehensiveness of their scope, the 
ancient method fell into comparative neglect among the 
mathematicians of the Continent with few exceptions, but 
as a method of discipline it was encouraged somewhat ex- 
tensively in the English schools. Among the more illus- 
trious names who maintained the excellence of the ancient 
Greek geometry may be mentioned Huygens (1629-1695) 
and La Hire (1640-1718) on the Continent, and Newton in 
England. To La Hire we owe the discovery of the theory 
of poles and polars. The discoveries of Leibnitz and New- 
ton in the infinitesimal calculus diverted attention for a 
time from pure geometry. Newton showed, however, that 
it could be employed in the higher branches of investiga- 
tion. Two English mathematicians, Cotes and Maclaurin, 
applied their methods to the investigation of geometrical 
curves. The astronomer Halley (1656-1742) by his beau- 
tiful translations of Apollonius, and Eobert Simson in his 
writings on the conic sections and prisms, endeavored to 
revive a taste for the ancient geometry; but their efforts 
were only instrumental in keeping alive the interest among 
a limited number of analysts in Great Britain and on the 
Continent. Little progress had been made till the brilliant 
discoveries of Monge and Carnot at the beginning of the 
present century. Gaspard Monge, the creator of descrip- 
tive geometry, was born at Beaune, France, in 1746, and 
died in 1818. His first edition of descriptive geometry 
was published in 1795 ; he also left another important 
work — " Application of Analysis to Geometry." His dis- 
coveries mark an epoch in the science of geometry, for 
which he did more than any other writer since Archimedes. 



202 GEOMETBY. 

Carnot, seven years younger than Monge, and a pupil of 
his, enriched the science by his " Geometry of Position" and 
his " Essay on Transversals." Mongers deductions showed 
the intimate relations between plane figures and figures in 
space; or that from the properties of bodies of two dimen- 
sions corresponding properties of bodies of three dimen- 
sions could be deduced. These relations gave rise to many 
new and elegant theorems. Carnotfs discoveries reached 
similar conclusions by pure geometry that Descartes had 
obtained by the analytical method. Arriving at the same 
results by different processes of investigation still further 
illustrated the vast possibilities of mathematical research. 
To the splendid discoveries of Monge and Carnot should 
be added those of Poncelet (1788-1867) in his remarkable 
treatise on " The Properties of the Projection of Figures," 
in which he employed the principle of continuity and the 
beautiful theorem of reciprocal polars and of homologous 
figures to demonstrate all known properties of lines and 
surfaces of the second order. 

This brief sketch will close with a reference to the great 
works of Chasles (1793-1880) on higher geometry, his 
treatise on prisms, researches on the attraction of ellipsoids, 
cones of the second order, ruled surfaces, and a memoir on 
duality and homography; a new method of determining 
the characteristics of systems of conies, and other produc- 
tions of this eminent master. The investigations in geom- 
etry are still progressing under the keenest analytical skill 
of two continents, and no one yet has fixed a limit — thus 
far and no farther. 

Elementary geometry in this country is based upon 
"Euclid's Elements" as translated and modified by Dr. 
Robert Sim son and improved by subsequent writers, or 
upon the treatise by Legendre, published first in 1794; 
but Legendre's text has been greatly improved by different 
editors. However, the most complete work that I have 



PBUIAEY COXCEPTIOXS. MJ3 

ever examined is " Trai aetrie Elementaire, par 

Eugdne Rouche et Cli. Pe Coinbeivusse." Paris, 1S68. 
This great work is divided into two parts. — PI t Gt 'dry 
and Oa :. Plane Geometry fills 328 pages. 

ancl Geometry in Space 472 page 

Pof a full and complete p: Son of the life and writ- 

ings of Euclid, and a history of the translations of his 
•'• Elements/*' the reader is referred to the article ,, 'Eu- 
cleides of Alexandria." by Professor Augustus De Morgan, 
Vol. II.. Smith's Dictionary of Greek and Soman Biogra- 
phy and Mythology. 



Teaching Geometry. 

Primary Conceptions. 

The final object in teaching geometry is to make good 

:is, sharply cut and accurately 
defined, must be i anted in the learner's mind. 

From the beginning he must know what he knows, and he 
must know what he does not know. It is assum t 

ner has mastered arithmetic and enough of algebra 
enable him ft i geometry intelli- 

tly. For those teachers who are desirous of starting 
younger pupils in geometry, the Primary Treatises by 
;ncer and Dr. Thomas Hill are the best, per- 
haps, published. However, the child of average ability 
has picked up. in oneway or another, considerable geo- 
metrical knowledge before he begins the subject in earnest. 
His knowledge is crude and unorganized. 

In teaching classes, or a single pupil, the first thing is to 

.r conception of a limited p f space. A 

crayon -box will answer the purpose well. It is held before 

the class. All see it. and can tell how many sides, ends, 



204 GEOMETRY. 

corners, and edges it has. They see, too, that it occupies 
a definite portion of space. Let the teacher remove it, 
and see if the members of the class can think of the defi- 
nite portion of space that the box formerly occupied. This 
brings the class to the conception of a portion of pure 
space, and from which the box has been removed. They 
need not follow the track of the box through space yet 
from one place to another; but their minds should be held 
to the contemplation of the " box-space" until it becomes 
a permanent notion. Now, let the box be brought before 
the class again. Tell the class that the top and bottom 
will remain the same size; but that the two side-pieces 
and end -pieces will gradually and evenly shrink away un- 
til the top and bottom approach each other and become 
one thin piece perfectly smooth and level. Question the 
class on this form. Next, assume that the ends begin to 
approach each other until they coincide, and question as 
before. If these changes in space can be readily followed 
by the class, assume the regular contraction of the sides 
and ends of the box at the same instant till they vanish. 
Upon the other hand, let the box begin to expand regu- 
larly till its length is ten feet, and the ends are enlarged 
proportionately. This last space may now be discussed by 
the members of the class. If necessary, the sides and ends 
may be assumed as having almost no thickness, and then 
contracted or expanded at pleasure, and the class ques- 
tioned till the conception of a " hedged-in portion of 
space" within space is a reality in the mind. Instead of 
an oblong box, a cubical block may be taken at first, or 
any other convenient object. The object to be accom- 
plished through illustrations is that of pure form; or in 
other words, the notion of the space a body occupies with- 
out respect to the material composing the body itself. 
This conception is the true starting-point in geometry, and 



PRIMARY CONCEPTIONS. 205 

it should be gained first through sensuous forms. The 
time required varies for different learners. 

The next step in the process is to take a definite surface, 
say, the top of the crayon-box, and let the class conceive 
the ends to contract regularly, the length of the sides 
remaining unchanged, until the two sides become one 
straight line; that is, the surface changes by contraction 
into a straight line. Use other illustrations, if necessary, 
to fix this notion. From the material line, however made, 
the process is to be continued until a conception of a pure 
geometrical line, as one of the boundaries of a pure solid, 
is obtained. 

Appropriate questions will enable the teacher to deter- 
mine this fact, and to correct any erroneous impressions 
the learner may have on the subject. A plane surface thus 
changes into a straight line, and by thinning the surface 
it is made to approach a pure geometrical line. By still 
further refining, the material line approaches, in thought, 
indefinitely near to the pure line in space with which 
geometry deals. Since the notion of a line can be thus 
derived from the surface or lid of the box, the line may 
be considered as shortening till it becomes a point. The 
successive steps are solid, surface, line, and point, as ob- 
tained through the material forms. Putting aside these 
crude notions, the pupil or the class must now pass to the 
space conceptions of the solid, surface, line, and point. 
The geometrical solid in pure space can now be considered 
without regard to the material form from which it was 
derived. 

Two of these surfaces that intersect form a line, and 
when three of them form a corner, it is a point. 

It is necessary for the pupil to obtain clear conceptions 
of a geometrical solid, surface, line, &n& point at the out- 
set. 

This process, as thus indicated, is essentially analytic; it 



206 GEOMETRY. 

needs to be reversed before the pupil apprehends it in all 
its force and beauty. Suppose he now assumes a geometri- 
cal point. This point is supposed to move in space. Its 
path is a line. If it move in one direction, its path is a 
straight line; hence from a point a line is generated. 
Suppose this line straight and ten inches in length, and it 
moves ten inches parallel to itself, thus describing a sur- 
face whose area is a hundred square inches. It is readily 
seen that a line can be moved in space so as to describe a 
surface. Next move the surface, or plane, ten inches par- 
allel to itself, and form a cube; hence a surface may be 
made to describe a solid. 

Geometry proper is the science of position, extent, and 
form; or more concisely, it deals with form abstractly. 

All material bodies occupy a limited portion of space; 
hence they have position, extent, and form; i.e., each body 
is someiohere, has some size, and is of some shape. 

To be Illustrated by the Pupil. 

Show that a solid has length, breadth, and thickness. 
Show that a plane surface has length and breadth. What 
are the edges of a page of writing-paper ? If two pieces of 
pasteboard cut each other, what does their intersection rep- 
resent? If two lines intersect, what do they form ? In 
a cube, how many faces meet in an edge ? In a corner ? 
Show that the path of a moving point is a line. Show that 
the path of a moving line is a surface. When is it not a 
surface? Why? When is the path of amoving surface a 
solid? What exception is there? Why ? 

Given a point; derive from it a line, a surface, a solid. 
Given a solid; reduce it to a point. What are the limits of 
a solid ? The limits of a surface ? The limits of a line ? 

These questions are merely suggestive; but they indicate 
the plan to be pursued. 



DEFINITIONS-EXPLAFA TIONS-POSTVLA TES. 207 



Definitions. 

Mathematics, as an abstract science,, rests substantially 
upon a few well-digested definitions. The key to geometry 
lies in its definitions. These must be mastered at the 
beginning; otherwise successful progress is impossible. 

1. A solid is extension haying length, breadth, and thick- 
ness. 

2. A surface is extension haying length and breadth. 

3. A line is extension haying length. 

4. A point is position only. 

Explanations. 

5. A Solid may be regarded under two conditions: 1. As 
a path formed by a moving plane; 2. As extension, having 
length, breadth, and thickness. 

6. A Surface may be regarded under three conditions: 
1. As the limit of a solid; 2. As the path formed by a 
moving line; 3. As extension, having length and breadth. 

7. A Line may be regarded under five conditions: 1. As 
a limit of a surface; 2. The intersection of two surfaces: 
3. The path of a moving point; 4. As the assemblage of 
all the positions of a generating point; 5. As extension in 
the direction of length. 

8. A point may be regarded under three conditions: 
1. As one of the limits of a line; 2. The intersection of 
two lines; 3. As position only. 

Postulates. 

1. A magnitude can have any position. 

2. A magnitude can have any form. 

3. A magnitude can have any extent. 



208 



GEOMETRY. 



Classification of Lines. 

1. Straight Lines, as : A 

2. Brokeu Lines, as : Z. 3 5 6 



2 



3. Curved Lines, as : 

4. Mixed Lines, as : E_ 



_3 5_ 

i 



Classification of Plane Surfaces. 



1. Rectilinear Surfaces, as: A 



" V 





ar 



2. Curvilinear Surfaces, as: (c) C^) 

B 

3. A Mixed Surface, as: A<^j>c 



Etc. 



Classification of Solids. 



1. Solids bounded by plane surfaces, as : 



2. Solids bounded by curved surfaces, as 



3. Solids bounded by mixed surfaces, as 




GENEBAL DEFINITIONS. 209 



General Definitions. 

1. An Axiom is a self-evident truth. 

2. An Absurdity is a self-evident falsity. 

3. A Postulate is a self-evident possibility. It states that 
something can be done, but does not tell how. 

4. A Theorem is a truth to be proved. 

5. A Problem is a question prepared for solution. 

6. A Proposition is the expression of a judgment. 

7. Theorems, problems, axioms, and postulates are called 
propositions. 

8. A Formula is a theorem expressed in algebraic lan- 
guage. 

9. A Corollary is an obvious truth deduced from the 
proposition to which it is attached. 

10. A Scholium is a remark upon some part of a propo- 
sition. 

11. A Lemma is an auxiliary theorem to be used in the 
demonstration of another proposition. 

12. A Demonstration is proof of a proposition. 

13. Demonstrations are of two kinds: 1. Direct; 2. 
Indirect. 

14. A Direct Demonstration proves a proposition in either 
of two ways: 1. By Superposition ; 2. By logical combina- 
tion of definitions, axioms, and previously demonstrated 
propositions. 

15. An Indirect Demonstration, called also Recluctio ad 
absurdum, proves a proposition true by showing that the 
supposition that it is false involves an absurdity. 

16. The Converse of a categorical or disjunctive proposi- 
tion is obtained by interchanging the subject and predi- 
cate. Thus, No A is B; conversely, No B is A. Also, A 
is B or C; conversely, B or C is A. 

17. The Converse of a hypothetical proposition is ob- 



210 GEOMETRY. 

tained by making the hypothesis of the original proposition 
the conclusion, and the conclusion the hypothesis. Thus, 
if A is B, it is not C,; conversely, if A is C, it is not B. 

18. The Solution of problems or exercises in Geometry 
consists of four parts: 1. The Analysis or course of thought 
by which the construction is found out. 2. The Construc- 
tion of the figure with the aid of compass and ruler. 3. 
The proof. 4. The discussion or limitations of the prob- 
lem. 

19. Similar Magnitudes are those that have the same 
form. 

20. Homologous points, lines, or surfaces, are similarly 
situated points, lines, or surfaces in similar magnitudes. 

21. Equivalent Magnitudes are those that have the same 
extent. 

22. Equal magnitudes are such that one can be applied 
to the other, and they coincide. 

23. Superposition is to apply mentally one magnitude to 
another. 

Questions for Review. 

1. What is meant by a Definition ? 

2. When is a definition redundant? When deficient? 

3. What is the literal meaning of the word definition ? 
From what word is it derived ? 

4. From what language is the word Axiom derived ? 
Absurdity? Postulate? Theorem? Problem? Propo- 
sition ? Judgment ? Expression ? Formula ? Corollary ? 
Scholium ? Lemma ? Demonstration ? Superposition ? 
Hypothesis ? Categorical ? Hypothetical ? Solution ? 
Magnitude ? 

Show in what respects each of the foregoing words varies 
from its original meaning. 

Remark. Pupils should be taught how to find these 



LINES AND ANGLES. 211 

words, and all similar terms, in the Unabridged Dictionary, 
and to trace out their meaning either from the Latin or the 
Greek. If the pupils have not learned the Greek Alphabet, 
they can do so in an hour or two. A severe drill in " Ety- 
mology," when Latin and Greek have been omitted, will 
enable pupils to resolve words with ease and pleasure. A 
word should be resolved into its etymological elements. 
For instance, the word " surface," derived from the Latin 
super, above, and fades, a face, has a different meaning in 
Geometry from the same word applied to a region of coun- 
try. 

Lines and Angles. 

1. Parallel lines are straight lines everywhere equally 
distant. They lie in the same plane. 

2. An angle is the difference in direction of two intersect- 
ing lines or of two intersecting planes. 

BA C is a plane angle; AB and A are the sides, and the 
point A is the vertex. The difference in direction of the 
two sides is the magnitude of the angle A. 

3. Angles are divided into two classes: 1. Right; 2. 
Oblique, and the oblique into the Acute and Obtuse. 

4. An angle is measured by the space through which one 
of the sides must turn in order to concide with the other 
side. 

While one side remains fixed in the plane, the other may 
revolve once around the vertex in the plane, thus forming 
an angle of 3G0 degrees; or twice, forming an angle of 720 
degrees; and so on: but if in one direction the angle is re- 
garded as positive, in the opposite direction it will be nega- 
tive. 

Euclid permits the student of Geometry to use a ruler 
and a pair of compasses to construct the propositions to be 
demonstrated. Instead, the pupil can use a straight-edged 



212 GEOMETRY. 

ruler, a string, and a piece of crayon for blackboard work, 
and a ruler and a circle-pen for paper diagrams. 

Postulates. 

1. A straight line can be drawn from any point, in any 
direction, to any extent. 

2. A straight line can be drawn from any point to any 
other point. 

3. A straight line passing through two points is fixed 
in position. 

4. A straight line can be produced indefinitely in either 
direction and to any extent. 

5. A circumference can be described from any center 
and with any radius. 

Suggestions. 

The definitions and explanations have been stated in 
full, and for the reason that the learner must lay a good 
foundation to succeed in the prosecution of the subject, 
Each technical term conveys a distinct idea, and these are 
the tools that the learner employs in his work; hence the 
importance of critically examining each, and ascertaining 
that it is in proper condition for immediate and successful 
use. To learn Geometry w T ell means persistent, intelligent, 
and severe mental application. The subject may be taught 
in such a manner as to blunt the intellects of the pupils; or, 
on the other hand, to stimulate them to the very highest 
degree of intellectual enthusiasm. 

The conditions for successful work are (1) active, ener- 
getic, and wide-awake pupils; (2) a good text-book with 
plenty of exercises for pupils; (3) ruler, string and crayon, 
and blackboard surface; or, pencil, ruler, circle-pen, and 
paper; (4) a first-class teacher, who understands the subject 
and can put spirit into the work. 



RECTILINEAR FIGURES. 213 

Rectilinear Figures. 

Under Book I. the following classification is adopted: 

1. Definitions and general principles.. 

2. Perpendiculars and oblique lines. 

3. Parallel lines. 

4. Triangles. 

5. Quadrilaterals. 

6. Polygons in general. 

7. Exercises. 

Definitions and General Principles. 

Whatever definitions are placed at the beginning of a 
book in Geometry are to be learned, illustrated, and assimi- 
lated. Committing definitions without understanding their 
meaning is of no educational or mathematical value what- 
ever, except in those special cases in which the learner is 
blessed with a very retentive memory, which will hold them 
long enough for actual knowledge to be acquired and classi- 
fied under the definitions. Under this head it is important 
that the teacher should train his pupils to give the three 
forms of definitions: — 1. Nominal or root definition; 2. 
Real definition ; 3. Genetic definition. 

When a term is used in a technical sense, this distinction 
should also be noted, and the two meanings compared. 

Perpendicular and Oblique Lines. 

When the pupil is called upon to demonstrate a theorem, 
(1) he must state exactly what he proposes to do ; in other 
words, lie states the question; (2) he proceeds to prove what 
he has said; (3) he must confine himself strictly to the sub- 
ject; (4) he must decide whether he has proved what he 
undertook; (5) he must resolve the conclusion back to prima- 
ry conceptions. The pupil should have within himself the 



2U 



GEOMETRY. 



means of testing his own work. Geometry as usually taught 
is barren of results, because the learner does not apply what 
he learns, and consequently it is soon forgotten. As an illus- 
tration, suppose the pupil is required to demonstrate that — 

11 At a given point in a straight line one perpendicular to 
the line can he drawn, and but one." 

After the proposition is assigned, the pupil will construct 
the figure. Next, he formally states what is to be proved. 
Then he proceeds with the proof. This concluded, he dis- 
cusses all possible limitations that can be imposed upon the 
question. Original demonstrations, corollaries, and remarks 
add sprightliness to all exercises. 



Steps. 




1. Construction. In the annexed diagram let AB be the 
straight line, P the given point, CP 
the perpendicular, and PD any other 
oblique line to AB at the point 0. 

2. Demonstration. Suppose AP to 
remain immovable while the line BP 
begins to revolve in the plane of the 
B paper about the point P as an axis. 
When the point B has reached D, the 
angle BPD is increasing and the angle APD is decreasing; 
consequently there is one point at which the angle BPD is 
equal to the angle APD. For the angle BPD was zero 
when PD coincided with PB, and when PD passes over 
to AP, then the angle APD is zero; hence there is one 
position, as CP, when the two angles are equal. 

3. Conclusion. " At a given point/ etc." 

4. Discussion. The theorem is not properly restricted. 
An infinite number of perpendiculars can be drawn from 
a point in a straight line. Suppose the perpendicular CP 



RECTILINEAR FIGURES. 215 

to revolve about the point P as an axis, and in every position 
being perpendicular to AB, it is evident that an indefinite 
number of perpendiculars can be drawn from a point in a 
straight line thus revolving. To limit the theorem properly, 
should be added " in a plane embracing the line." 

5. Questions. If the angle BPD — 60°, what is the angle 
DPC ? What is the complement of the angle BPD ? Of 
the angle CPD ? What is the supplement of the angle BBC? 
What is the angle that the line AP makes with the line 
BP at the point P ? Prove that the right angle APC is 
equal to the right angle BPC. Prove that the complements 
of equal angles are equal. That the supplements of equal 
angles are equal. If the angle APD is equal to 115° 30', 
what are the values of the angles BPD and DP (7 respectively? 
How r many degrees are there in the complement, and in the 
supplement, of -f of a right angle ? How many degrees are 
there in an angle whose complement is five times the angle ? 
If the supplement of an angle is five times the angle, w r hat 
is the complement of the angle? How many degrees are 
there in the angles BPD + DPC + CPA ? 

Another proposition will be given to indicate still further 
the method of teaching. 

Theoeem. The two adjacent angles which one straight 
line makes with another are together equal to two right 
angles. 

1. Construction: In the diagram A B and P C are the 
two straight lines which meet at the point P. 

2. If the angle APC is equal to the angle BPC, then the 
two angles are right angles, and are equal by definition, 
and CP will coincide with DP. 

If they are not equal, let DP be drawn perpendicular to 
AB, and then the angle APD is equal to the angle BPC 
plus the angle CPD. 

But the angle BP C plus the angle CPD are equal to a 
right angle; hence the angle APD plus the angle DPC 



216 



GEOMETRY, 



plus the angle CPD are equal to two right angles, because 
the angle DPC plus the angle CPB are equal to the angle 
DPB. 

1. Corollary. The sum of all the consecutive angles 
formed by any number of straight lines in the same plane, 
drawn from the same point of a straight line on the same 
side, is equal to two right angles. Prove. 

2. When the sum of the successive angles at a point in a 
line is equal to two right angles, the two extreme arms form 
a straight line. 

3. The sum of all the angles formed by straight lines in a 

plane, meeting at a common 
point, is equal to four right 
angles. Prove from the an- 
nexed figure. 

If Z APC = 27° 30', 
Z CPD = 36°, IDPE = 
38° 30', zFPB = ±r4t5'; 
how many degrees in the 

Z BPF? 
If Z APG = 82° 29' 30", how many degrees in the 

Z GPB? 

4. Angles may be added or subtracted. Illustrate. 

5. The greater an angle the less its supplement. 

6. Vertical or opposite angles are equal. 

Prove from the annexed figure that the angle APC is 
equal to the angle BPD ; also, that the angle APD is the 
supplement of either the angle APG or the angle BPD. 

Definitions that are Constantly Used. 

The sooner the learner familiarizes himself with the fol- 
lowing geometrical terms and their meaning, the more in- 
telligent and satisfactory will be his progress. He must 
accustom himself to think in geometrical language. 

1. Exterior and Interior Angles. 

2. Exterior angles of a Triangle, 




PARALLEL LINES AND ANGLES. 217 

3. Alternate Angles. 

4. Corresponding Angles. 

5. A Transversal. Angles formed by a Transversal. 

6. Bisector of an angle. Bisectors of vertical angles. 
Bisectors of a pair of vertical angles formed by two inter- 
secting straight lines. 

7. Bisector of an angle as a Locus. 

Special importance should be attached to the hypothesis, 
demonstration, and conclusion of each proposition. Then 
if the proposition admits of a converse proposition, that also 
should be enunciated. By all means the learner must start 
right, must keep right, must learn thoroughly, and be able 
to give a valid reason for every step he takes. 

As new terms are introduced, their etymological meaning 
should be critically investigated. Words are the instru- 
ments of thought. 

Parallel Lines and Angles. 

1. Two straight lines in one plane may be discussed 
under the following conditions : 

(a) If they are parallel, they never meet. 

(b) If they can not meet, they are parallel. 

(c) If they can meet, they are not parallel. 

(d) If they are not parallel, they can meet. 

(e) If they intersect, they form oblique or right angles. 

(f) If one angle is oblique, all the other angles are 
oblique. What exception ? 

(g) If one angle is right, all the other angles are right. 

What exception ? 

2. When two or more straight lines intersect in two or 
more points, but not forming an enclosed figure, the follow- 
ing cases deserve special notice: 

(a) When two or more straight lines are perpendicular to 
the same straight line, and conversely. 



218 GEOMETRY. 

(5) When two parallel straight lines are cut by another 
straight line the alternate interior angles are equal ; the 
alternate exterior angles are equal ; the corresponding angles 
are equal ; and the interior angles on the same side of the 
secant line is a constant quantity, equal to 180 degrees. 

The converse of these conditions in each case is true. 

(c) Two angles whose sides are parallel each to each, are 
either equal or supplementary. Note the cases when the 
parallel sides extend in the same direction, or in opposite 
directions. 

(d) Two angles whose sides are perpendicular each to 
each are either equal or supplementary. Note the cases if 
both angles are acute or both obtuse; if one is acute and 
the other obtuse. 

The doctrine of parallel lines is a very important one. 
Parallel lines in Geometry may beregarded a s akind of 
ivheelbarroiv, upon which almost everything can be loaded 
and pushed along. Important propositions will be strongly 
accentuated. 

Questions. Two straight lines intersect; one of the 
angles is 83° 44' 33": what are the other angles? Two 
parallel lines are cut by a third line; if one angle is 36°, 
what are the other seven angles ? 

Two straight lines meet in a point; they are cut by a 
third line; if one angle is known, formed by the third line 
and either one of the other lines, how many other angles 
can be found ? 

Illustrate each question by a diagram. 

Demonstrations. 

Demonstrations different from the ones employed in the 
text should be discovered by the pupils. Such demonstra- 
tions are more lasting, and make deeper impressions on the 
mind. If a demonstration is to be written out, it should 
be a model of neatness in arrangement and of logical ac- 




PABALLEL LINES AND ANGLES. 219 

curacy in reasoning. In no other species of composition is 
there so great necessity for precision in the use of language. 
As an illustration, suppose this familiar theorem: " If two 
parallel lines are cut by a third straight line, the corre- 
sponding angles are equal." 
The construction is omitted. 

1. Demonstration. Suppose the lines CD and EF to re- 
main fixed, and let the line AB be 
moved parallel to itself toward CD 
till it coincides with the line CD, 
then the point A falls on C, G on 
H, andi?onD; therefore the angle 
HGB coincides with the angle EHD, 
the angle A GF with the angle CHG, 
the angle A GH with the angle CHE, 
and the angle FGB with the angle 
GHD, — which prove that the corresponding angles "are 
equal. 

2. Demonstration. Conceive the angles at G transferred 
to H, the direction of the lines remaining unchanged; then 
each angle will coincide with its corresponding angle, and 
be equal to it. 

Remark. Theorems may often be arranged in groups of 
four: 1. The original theorem; 2. Its opposite; 3. Its con- 
verse; 4. The converse of the opposite. 

Example. "1. If two lines are parallel, the correspond- 
ing angles will be equal. 2. If two lines are not parallel, 
the corresponding angles will be unequal. 3. If the corre- 
sponding angles are equal, the lines will be parallel. 4. If 
the corresponding angles are unequal, the lines will not 
be parallel." 

Exercise on Lines and Angles. 

The test of the learner's knowledge is his ability to handle 
successfully original exercises. Under each theorem exer- 
cises should be given for practice. 



220 GEOMETRY. 

1. Five straight lines in a plane meet at a point, making 
equal angles with one another around that point: how 
many degrees in each angle, and what part of a right angle 
is each angle ? 

2. Prove that the bisectors of adjacent supplementary 
angles are at right angles to each other. 

3. Find the angle between the bisectors of adjacent com- 
plementary angles. 

4. Ten lines meet at a point so as to form a regular, ten- 
rayed star : what is the angle in degrees between two con- 
secutive rays ? 

5. If A is the number of degrees in any angle, prove that 
90°+ A° is the supplement of 90°- A°; and that 45°+ A° 
is the complement of 45°— A°. 

Triangles. 

The following terms are of frequent use, and if not 
already known, are now to be learned : 

1. The kinds of triangles; as, Plane, Acute, Oblique, 
Eight, Isosceles, Scalene, Equilateral. 

2. The parts of triangles; as, Sides, Base, Altitude, Ver- 
tex, Perpendicular, Hypothenuse, Legs, Median, Perimeter, 
Bisectors, Area. 

3. Interior Angles, Exterior Angles, Vertical Angles, 
Base Angles. 

4. Terms of Comparison; as, Homologous sides, Homol- 
ogous angles, Similar triangles, Equal angles, Equal areas, 
Equivalent areas. 

Let the learner give the derivation of Isosceles, Scalene, 
Equilateral, Altitude, Vertex, Perpendicular, Hypothe- 
nuse, Median or Medial, Perimeter, Area, Vertical, Homol- 
ogous, and Similar. 

Important Properties of Triangles. 
Such properties as are deemed most important for the 
pupil to know will be mentioned. 



TRIANGLES. 221 

1. The sum of the angles of a triangle is equal to a con- 
stant quantity, i.e., two right angles. 

This theorem should be proved in two different ways. 
The simplest proof is by drawing a line through the vertex 
of the triangle parallel to the base. 

From this theorem also follows: 1. Every angle of a 
triangle is the supplement of the sum of the^ other two. 

2. If one side of a triangle is produced, the exterior angle 
is equal to the sum of the two opposite interior angles. 

3. That in every triangle at least two of the angles are acute. 

4. If two of the angles of a triangle are equal, they are both 
acute. 5. In a right-angled triangle the two acute angles 
are complementary. 

2. Each side of a triangle is less than the sum of the other 
two, and greater than their difference. 

3. The angle contained by two straight lines drawn from 
any point within a triangle to the extremities of one of the 
sides is greater than the angle contained by the other twc 
sides of the triangle. 

4. Conditions of Equality. 

(1) When the three sides of one triangle are respectively 
equal to the three sides of the other. 

(2) When two sides and the included angle of the one are 
respectively equal to the two sides and included angle of the 
other. 

(3) When one side and two angles of the one are respec- 
tively equal to the corresponding elements of the other. 

(4) Two triangles are equal when one of them has two sides, 
and the angle opposite to the side which is not less than the 
other given side respectively equal to the corresponding 
elements of the other triangle. 

This is the ambiguous case which occurs occasionally in 
the construction of problems in geometry, and in the solu- 
tion of certain trigonometrical problems. This case is a 
stumbling-block to the learner unless it is thoroughly 



222 



GEOMETRY. 



mastered. Perhaps the best way to impress it upon the 
learner's mind is to bring out what may be regarded as the 
exceptions to the general statement, namely, Three Elements 
are enough to determine a plane triangle. By Elements is 
meant the three sides and the three angles. 



Two unequal tri- 




Exceptions. 

1. When the three angles are giyen. 
angles may haye their angles equal. 

2. When two unequal sides and the angle opposite to the 
less side are giyen. There are two triangles that fulfill the 

conditions. Thus, if AB is greater 
than BC, and the angle BAC is 
given, it is evident that the tri- 
angle ABC or ABD will satisfy 
the conditions; hence the ambig- 
uity. This exceptional case should be fully and thoroughly 
discussed before the pupil passes over it. 

Conditions of Equality in Right Triangles. 

1. Two right triangles are equal when the hypothenuse 
and a side of one are respectively equal to the hypothenuse 
and a side of the other. 

2. When the hypothenuse and acute angle of the one are 
equal to the hypothenuse and acute angle of the other. 

3. When aside adjacent to the right angle and the adjacent 
or opposite acute angle are respectively equal to the corre- 
sponding elements in another right triangle, they are equal. 



Conditions of Inequality. 

Here, again, the principle of contrast is employed to give 
effect to the teaching as well as in the method of retaining 
what is already learned. Both principles — equality and 
inequality — are used so frequently that the learner can 
hardly be too familiar with them. Let the learner state 



TBIANGLES. 223 

each condition of equality, and then its converse; also its 
opposite. Very much of geometrical reasoning is carried 
on by the aid of equal triangles and similar triangles. 

Illustrative Questions. 

1. State all the conditions of equality in two plane tri- 
angles. 

2. Of two right triangles. 

3. Two equal lines, a and b, are joined to the line c; 
show that they make equal angles with c. 

4. The bisector of the vertical angle of an isosceles tri- 
angle bisects the base; show that the two triangles thus 
formed are equal right triangles. 

5. ABC is an isosceles triangle, and the angle A is twice 
either B or C; prove that A is a right angle. 

6. If A is half of either B or C, how many degrees does 
it contain ? How many degrees are there in the angle C? 

7. The vertical angle of an isosceles triangle is 36 degrees. 
How many degrees in each base angle ? If the sides and 
base be produced, determine all the exterior angles. 

8. Prove that the bisectors of the equal base angles of 
an isosceles triangle form with the base another isosceles 
triangle. 

Remark. Questions on a proposition may very appropri- 
ately be divided into three classes : 1. Those on the proposi- 
tion itself, including agreements and differences with other 
propositions; 2. Numerical . applications connected with 
the proposition; 3. Geometrical exercises. The last in- 
cludes original problems. The order in the mental process 
appears to be: 1. Learning the new; 2. Connecting it Avith 
what is previously known; 3. Applying the knowledge to 
practical and theoretical problems. 

Geometry is learned to be used. 

The more important propositions will be dwelt upon, 



224 GEOMETRY. 

that the reader may get the methods of teaching rather 
than a minute analysis of every theorem in an elementary 
treatise on geometry. 

Important Theorems. 
The following theorems are generally recognized as very 
important owing to their wide application: 

1. Every point in the bisector of an angle is equally dis- 
tant from the sides of the angle. 

Let the pupil give the converse of this theorem. 

Remark. From a discussion of these two theorems a very 
correct notion of what is meant by the word Locus in its 
simplest sense is obtained. 

2. The three bisectors of the three angles of a plane tri- 
angle meet in a point. 

3. The three perpendiculars erected at the middle points 
of the side of a triangle meet in the same point. 

4. The three perpendiculars from the vertices of a triangle 
to the opposite sides meet in the same point. 

5. The three medial lines of a triangle meet in the same 
point. 

These theorems are among the most important in Book I. 
The fifth deserves special attention owing to its peculiar 
properties. The intersection of the three medials is the 
center of gravity of the triangle — a truth worth remember- 
ing. The geometrical demonstration is very pretty. 

Questions. 

1. A line is perpendicular to another line at its middle 
point ; show that it is the locus of the points equally distant 
from the extremities of the line. 

2. Is the bisector of the vertical angle of an isosceles tri- 
angle a locus ? Illustrate. 

3. If the two equal sides of an isosceles triangle decrease 
uniformly till the area of the triangle is zero, what kind of 




QUADRILATERALS. 225 

a line will the vertical angle describe ? What is the line 
with respect to the base ? 

4. What is the locus of a point equally distant from hvo 
fixed points ? From two fixed parallel lines ? From two 
intersecting straight lines ? 

Quadrilaterals. 

Quadrilateral is from the two Latin words, quatuor, four, 
and latus, side. It is, therefore, any four-sided polygon. 
The most important simple properties of quadrilaterals 
relating to their angles are: 

1. The sum of its interior angles are equal to four right 
angles. 

2. By substituting polygon for quadrilateral, the general 
law is expressed thus: The sum of all the angles of any poly- 
gon is equal to two right angles taken as many times less two 
as the polygon has sides. 

3. If each side of a polygon he produced in one direction, 
the sum of all the exterior angles is equal to four right 
angles. 

The learner should observe the particular case when one 
or more angles are re-entrant. 

Quadrilaterals are divided into classes as follows : 

1. The Trapezium; 2. Trapezoid; 3. Parallelograms. 

Parallelograms are divided into two species : 1. Rhomboid 
and Rhombus; 2. Rectangle and Square. 

The peculiar properties of each of these figures should 
be learned. They may be classed under the relative direc- 
tions of the sides, the nature of their angles, and the char- 
acter of their diagonals. 

It is from the parallelogram that the learner must get 
the first definite notion of finding exactly the area of a tri- 
angle. 

The standard of surfaces is the square, the unit of area. 
Parallelograms are determined when their bases and alti- 



226 GEOMETRY. 

tudes are known. Since a triangle is half a rectangle, hav- 
ing the same base and altitude, its area is also known. 
The product of two lines is a rectangle; but the product 
of two algebraic quantities may also be interpreted as a 
rectangle. The method of reasoning geometrically about 
magnitudes is called the ancient method, while the alge- 
braic method belongs to the discoveries of Descartes. 
Tlie measurement of areas may be classed as follows : 
1. Of Eectangles; 2. Of Parallelograms; 3. Of Triangles; 
4. Of Trapezoids and Trapeziums; 5. Of Polygons in gen- 
eral. 

Questions. 

1. A square is a parallelogram whose sides are all equal 
and whose angles are all equal. Is this definition defective 
or redundant ? Give reason for your answer. 

2. In what respects do a trapezoid, a rhombus, and a 
rectangle differ? In what respects do they agree? 

3. When does a parallelogram become a rectangle ? A 
rectangle a square ? 

4. What figure is at once a rhombus and a rectangle? 
Show how this can be. 

5. Which quadrilaterals have their diagonals unequal ? 
If the diagonals of a quadrilateral be equal, what must the 
quadrilateral be? 

6. If one angle of a parallelogram be a right angle, what 
must the other three angles be ? Does the definition of 
parallelogram include the preceding answer ? Why ? 

7. Is there a rule in arithmetic to find the diagonals of 
a rectangle if the side and end be given? What is the 
rule ? What principle does it involve ? 

A good collection of exercises should conclude each 
book in Geometry. These exercises are more valuable, if 
worked by the pupils, than the theorems which are demon- 
strated. 



THE CIRCLE. 227 

The Circle. 

The first element of geometry is the right line and the 
other element is the.circle, and all constructions that can 
be made by these two are regarded as strictly geometrical. 
However, the straight line and circle are employed in trigo- 
nometry, analytical geometry, and in other departments of 
the more advanced mathematics. 

In Elementary Geometry the more important properties 
of the circle may be arranged under the following subdivi- 
sions: 

1. General Definitions. 

2. Arcs and Chords. 

3. Tangents and Secants. 

4. Measurement of Angles. 

5. Relative Positions of two Circles. 

6. Exercises. 

General Definitions. 

The general definitions that must be accurately learned 
and retained are: 

1. Circle ; 2. Circumference ; 3. Diameter ; 4. Radius ; 
5. Center ; 6. Arc ; 7. Semi-circumference ; 8. Quadrant ; 
9. Chord ; 10. Segment ; 11. Sector; 12. Tangent ; 13. Point 
of Tangency: 14. Secant ; 15. Central Angle or Angle at 
the Center ; 16. Inscribed Angle ; 17. Inscribed Segment ; 
18. Inscribed Triangle ; 19. Inscribed Polygon ; 20. In- 
scriptible ; 21. Circumscribed ; 22. Escribed ; 23. Center 
of Symmetry ; 21. Axis of Symmetry ; 25. Symmetrical ; 
26. Conjugate Arcs; 27. Normal. 

Arcs and Chords. 
The principles regarded as essential which the learner 
should master will be enumerated under each subdivision. 
1. A straight line can cut a circumference in two points onlj\ 



228 GEOMETRY. 

2. Every point at a less distance from the center of a circle 
than the radius is within the circumference. 

8. Every point at a distance from the center equal to the 
radius is in the circumference. 

4. Every point at a distance from the center greater than 
the radius is without the circumference. 

5. Every diameter bisects the circumference, and also bi- 
sects the circle. 

6. In equal circles, or in the same circle, equal angles in- 
tercept equal arcs on the circumference, and if the arcs are 
equal, the angles are equal ; also in equal circles, or in the 
same circle, equal arcs are subtended by equal chords, and 
conversely. 

7. The greater arc is subtended by the greater chord, if the 
arc is less than a semi-circumference. 

8. If a diameter is perpendicular to a chord, it bisects the 
chord and the arcs subtended by it. 

9. Equal chords of equal arcs are equally distant from the 
center , and of two unequal chords in the same circle or in 
equal circles the less is farther from the center. The con- 
verse of these propositions is also true. 

10. Through three points not in the same straight line one 
circumference only can be drawn. 

Every proposition should be proved either by the method 
in the text or in an independent manner by the pupil. The 
teacher will bear in mind that the best training is that 
which fits the learner to be an original reasoner, and to de- 
cide upon the validity of his own processes of reasoning. 
Not a few teachers of Geometry have gone so far as to ex- 
clude the text entirely from the class, and instead to assign 
theorems and exercises for the pupils to demonstrate ac- 
cording to their own devices. 

Of the ten propositions under Arcs and Chords, it is 
evident that the fifth, sixth, seventh, eighth, ninth, and 



TANGENTS AND SECANTS. 229 

tenth require more careful investigation than the second, 
third, and fourth. 

At first the class need diagrams as aids, but later on 
these may in a great measure be dispensed with, except in 
rare instances, when the construction is difficult, or some 
obscure point is to be explained. 

Figures should be seen mentally, and a little practice 
will enable the learner to construct such and to hold them 
in the mind almost as clearly as when they are outlined on 
the board. 

Tangents and Secants. 

The most elementary notion yet developed in Geometry 
is that of the point, which has position only. From the 
point to a line another notion or idea is involved. The 
two simplest conditions under which a straight line can be 
drawn are that one point of the line and its directioji 
shall be know T n, or, in other words, two points determine 
the direction of a line. A circle is fully determined when 
its centre and radius are known. These two elementary 
notions determine the position and size of the circle. From 
these the circle can be drawn at once ; and the next sim- 
plest condition is when three points are given not in the 
same straight line. Likewise the simplest problem in Tan- 
gencies is when a straight line is given in position and a 
point out of the line, to draw a circumference tangent to 
the given line. However, the subject of Tangencies is one 
of the most varied and extended in its applications within 
the entire range of Elementary Geometry. 

As much as can be attempted in this connection is to call 
attention to a few of the fundamental principles underly- 
ing this interesting subject. 

1. If a straight line is oblique to a radius at its extrem- 
ity, it cuts the circumference ; hut if it is perpendicular to 



230 GEOMETRY. 

the radius at its extremity, it is tangent to the circle. The 

converse of these propositions is true. 

2. If a secant cutting the circumference is made to re- 
yolve about one of the points as a pivot, the secant becomes 
a tangent when the two points of intersection coincide. 

This proposition shows that the tangent is a special case 
of the secant. 

3. Two parallels intercept equal arcs on a circumference. 
There are three cases : 

1. When the parallels are both secants. 

2. When one of the parallels is a secant, and the other is 
a tangent. 

3. When both the parallels are tangents. 

4. Two tangents to a circumference, drawn from a point 
without, are equal, and make equal angles with the straight 
line joining the point with the center of the circle. 

Propositions third and fourth are the most important 
ones in this group. Yet important is used in a relative 
sense, and must not be interpreted as conveying the idea 
that the others are useless. 

Measurement of Angles. 

The right angle, or 90 degrees, is the natural unit of 
angular measurement when comparing one angle with 
another ; but in actual practice it has been found more 
convenient to divide the right angle into ninety equal parts, 
called degrees ; each degree into sixty equal parts, called 
minutes ; and each minute into sixty equal parts, called 
seconds. To every angle there corresponds an arc of a cir- 
cumference, and either the angle or its corresponding arc 
of a definite length may be taken as a medium of compar- 
ison with any other angle in the same or equal circles. 
Angles are compared in the same manner as other geo- 
metrical magnitudes. 



MEASUREMENT OF ANGLES. 231 

The principles underlying the measurement of angles 
are : 

1. A central angle is measured by its intercepted arc. 

2. A central angle is proportional to its intercepted arc. 
These principles give rise to the following theorems : 

1. An inscribed angle is measured by one half its inter- 
cepted arc. 

Case I. When one side of the angle is a diameter. 

Case II. When the center is within the angle. 

Case III. When the center is without the angle. 

Each of these cases should be mastered in detail. The 
demonstrations are very simple. 

In connection with these cases are associated the follow- 
ing deductions: 

1. All the angles inscribed in the same segment of a cir- 
cle are equal. 

2. All the angles inscribed in a semicircle are right 
angles. 

3. The opposite angles of an inscribed quadrilateral are 
supplementary. 

4. An angle at the center of a circle is double the angle 
at the circumference having the same arc. 

2. An angle formed by a tangent and a chord is measured 
by one half its intercepted arc. 

3. An angle formed by two chords intersecting within the 
circumference is measured by one half the sum of the arcs 
intercepted between its sides and between the sides of its 
vertical angles. 

4. An angle formed by two secants, or by a tangent and a 
secant, is measured by one half the difference of their inter- 
cepted arcs. 

5. If a quadrilateral circumscribes a circle, the sum of one 
pair of opposite sides is equal to the sum of the other pair. 



2X2 GEOMETRY. 

Relative Positions of two Circles. 

When two circles lie in the same plane, they occupy 
some one of the following positions with respect to each 
other. 

1. They may be wholly external. 

2. They may be tangent externally. 

3. They may intersect. 

4. They may be tangent internally. 

5. One may lie wholly within the other. 

6. They may be concentric. 

7. They may be coincident when equal and concentric. 

Conditions arising from Intersections. 

1. If two circumferences intersect, they cut each other in 
two points. 

2. The straight line joining the centers of two circles- 
bisects their common chord at right angles. 

3. The distance between the centers of two intersecting 
circles is less than the sum of their radii and greater than 
their difference. 

Conditions arising from Tangency and Non-Intersection. 

1. When two circumferences are tangent to each other 
externally, the distance between their centers is equal to 
the sum of their radii. 

2. When two circumferences are tangent internally, the 
distance between their centers is equal to the difference of 
their radii. 

3. When one circumference is wholly within the other, 
the distance between their centers is less than the differ- 
ence of their radii. 

If the pupil does not understand each of the important 
propositions enunciated, he should draw a diagram, and 
prove the relation required. He must study the relations 
of the lines, angles, arcs, and circumferences, and endeavor 



RELATIVE POSITION OP TWO CIRCLES. 233 

to discover from what is given how to determine what is 
required. 

Questions and Exercises Illustrating the Text. 

The following questions and exercises are designed to in- 
dicate partly the method to be pursued in conducting reci- 
tations. The pupil should be strong on the side of pres- 
entation as well as in comprehension of a subject. Inven- 
tion and arrangement are two desirable qualities of mind 
to be cultivated. 

1. Four equal chords are drawn in a circumference form- 
ing a -square : required the locus of the middle points of 
the chords. 

2. What is the difference between a diameter and a 
chord ? A semicircle and a segment ? A secant and a 
tangent ? 

3. A diameter of a circle is 20 inches, and a chord is 10 
inches ; how far are they apart if the chord is parallel to 
the diameter ? 

4. A system of chords are parallel in a circle ; what is 
the locus of their middle points? 

5. What theorems are involved in question 4 ? 

6. Any number of chords of a circle are drawn through 
a point on its circumference ; what is the locus of their 
middle points? Give all the principles involved in this 
proposition. 

7. Under what conditions will a tangent and a secant to 
the same circle not intersect ? 

8. When they do not intersect, how are they limited ? 
Give a reason for your answer. 

9. When does a secant become a chord ? A tangent a 
point ? 

10. Show that tangents to a circle at the extremities of 
a diameter are parallel. 



234 GEOMETRY. 

11. The arc of a chord is 135 degrees ; what part of the 
whole diameter is it ? 

12. The diameters of two circles are 10 and 12 inches 
respectively : 1. If the circles are tangent externally what 
is the distance between their centers ? 2. Tangent inter- 
nally? 3. When their centers are 30 inches apart, what 
is the distance between their circumferences? 



On the Construction of Problems. 

Formerly it was the custom of American authors to give 
problems in one book of their geometries, and the learner 
was required to reproduce the work which the author had 
already given, instead of making his own constructions and 
demonstrations. Book V. of a well-known Geometry, which 
the writer studied first, contained all the problems, — all of 
which were demonstrated,— and there appeared to be no 
necessity for anything additional to these thirty-two prob- 
lems in the text. A reaction set in, and now all American 
works of any value contain copious exercises. Had I to 
make a choice of a Geometry with no exercises, and one 
composed entirely of exercises and all else omitted, I would 
select the latter as preferable. There is an independent 
habit of thought derived from constructing and demon- 
strating problems that cannot be secured from a second- 
hand line of argument. All exercises are designed to test 
the learner's original power and skill. He fixes in his mind 
the given conditions. These he must hold firmly. Next 
he employs his constructive imagination in arranging the 
necessary diagram to represent not only the given condi- 
tions, but those that are required. Finally, step by step 
he proceeds to prove the truth or falsity of the proposition. 

The object of this kind of mental discipline is to teach 
the learner to know truth when he finds it. It being 



ON THE CONSTRUCTION OF PROBLEMS. 235 

found, he then sets limits to the truth arrived at under the 
conditions imposed by the problem itself. 

To construct and to demonstrate a problem involve six 
parts : 

1. To read the problem and to remember it. 

2. To draw the given parts. 

3. To draw the additional parts when necessary. 

4. To demonstrate it, i.e., to find the truth. 

5. To find its limits. 

6. To resolve it back into primary definitions. 

When the pupil first begins geometry the figures are 
assumed to be constructed, and they are used as helps in 
the demonstration of principles. Later on he learns how 
to construct figures solely by the aid of the straight line 
and circumference, or the ruler and string on the black- 
board, or the ruler and compasses on paper. 

The successful teacher of geometry will give exercises, if 
the text does not contain them, under each theorem, and 
there should also be a large collection of well-graded exer- 
cises at the end of each book. These exercises should em- 
brace theorems, geometrical loci, determinate problems, and 
algebra applied to the solution of geometrical questions. 
AVhenever possible, geometrical principles should be traced 
down through algebra and arithmetic to the simplest 
forms under which magnitudes are investigated. 

Let the fact be deeply impressed upon the learner's mind 
that every principle in geometry is put there for a purpose, 
and it is important that it should be learned, understood 
thoroughly, and retained. That there is no telling when it 
may be required in the demonstration of a theorem or in 
the solution of a problem, and that he is the best geometer 
who keeps his knowledge well in hand and can use any 
portion or all of it whenever the occasion demands it. 

The construction of problems ought not to be postponed 
till the pupil or class has studied two or three books ; but 



236 GEOMETRY. 

it should be carried along with the demonstration of the- 
orems from the start. The more constructing and proving 
and testing the learner does, the greater the interest and 
the more rapid will be his progress. 

Area and Equivalency. 

The word " area " is used in the sense of a numerical 
measure, and " equivalency" is applied to magnitudes hav- 
ing the same size, but not the same shape. The area of a 
surface is the number of square units required to cover it. 

Writers often use such expressions as "the rectangle of 
two lines, the product of two lines" — meaning thereby the 
product of their numerical measures instead of the lines 
themselves. 

The learner should master the following principles : 

1. Parallelograms having equal bases and equal altitudes 
are equal. 

2. Every triangle is one half of a parallelogram having 
the same base and the same altitude. 

3. Two rectangles having the same or equal bases are to 
each other as their altitudes, and conversely. 

This truth is expressed in general by saying that any two 
rectangles are to each other as the products of their bases 
by their altitudes. 

4. The area of any parallelogram is equal to the product 
of its base by its altitude ; and the area of any triangle is 
equal to the product of its base by half its altitude, or half 
its base by its altitude. 

5. The area of a trapezoid is equal to half the product of 
its altitude by the sum of its parallel sides. 

6. The square of the sum of two lines is equivalent to the 
sum of their squares plus twice their rectangle. 

7. The square of the difference of two lines is equivalent 
to the sum of their squares minus twice their rectangle. 



AREA AND EQUIVALENCY. 237 

8. The rectangle of the sum and difference of two lines is 
equivalent to the difference of their squares. 

Propositions 6, 7, 8 should be demonstrated geometri- 
cally as well as algebraically. They are the well-known 
algebraic theorems, numbered I, II, and III. They afford 
fine illustrations of comparing the two methods of reasoning. 

9. The square of the hypothenuse of a right triangle is 
equiyalent to the sum of the squares of the other two sides. 

Many demonstrations of this theorem have been given, 
but the 47th, in the first book of Euclid, is one of great 
elegance, and takes precedence over all others. The student 
should not only demonstrate it when the squares are de- 
scribed on the outer side of the triangle, but when they 
are described on the inner sides, or when one square is de- 
scribed on the outer side and two on the inner sides, or vice 
versa. 

Owing to the historic interest associated with this theo- 
rem, its great mathematical value, its wide application, 
and the numerous theorems dependent upon it, it is justly 
regarded as one of the most important theorems in geome- 
try. The statement of the theorem as given in this con- 
nection is a special case of a general statement. 

10. In any triangle the square of the side opposite an 
acute angle is equal to the sum of the squares of the other 
two sides, diminished by twice the product of one of these 
sides and the projection of the other side upon it. 

Discuss both cases. Let the learner compare the right- 
hand member of the equation with Theorem II, in algebra. 

11. In an obtuse-angled triangle the square of the side 
opposite the obtuse angle is equivalent to the sum of the 
squares of the other two sides, increased by twice the prod- 
uct of one of these sides and the projection of the other 
side upon it. 

Here the learner should compare the right-hand member 
of the equation with Theorem I. If A B is the side opposite 



238 GEOMETRY. 

the given angle, #(7 and AC the other two sides, and CD 
is the projection, then theorems 10 and 11 are expressed 
thus : 

AI? = DC 2 + AC* =F BC X CD. The minus sign is 
used when the angle is acute, and the plus sign when it is 
obtuse. 

These theorems have an important bearing in trigonom- 
etry. 

12. The sum of the squares of two sides of a triangle 
is equivalent to twice the square of the median line to the 
third side together with twice the square of half the third 
side. 

Remark. If instead of finding the equivalent of the sum 
of the squares of the two sides their difference be required, 
the expression is greatly simplified. 

Nos. 10, 11, 12 show how the values may be obtained by 
substitution without resorting to algebraic processes. That 
is, the values of lines usurp the functions of unknown 
quantities. Frequently it becomes necessary to eliminate 
and to substitute unknown quantities in a geometrical 
demonstration, and this fact, I am inclined to believe, had 
much to do in laying the foundation for our modern alge- 
braic methods. 

Another and perhaps a better illustration is found in 
the following theorem : 

13. In any quadrilateral the sum of the squares of the 
four sides is equal to the sum of the squares of the diagonals, 
increased by four times the square of the line joining the 
middle points of the diagonals. However, this proposition 
is little used in geometry, but it is good practice for the 
learner to make the reduction. If the length of the line 
joining the middle points of the diagonals is zero, a new 
phase of the problem is presented, which the learner will 
readily recognize. These unexpected turns add great in- 
terest to mathematical investigations. 






PROPORTIONALITY AND SIMILARITY. 239 

14. To find the area of a triangle when its three sides are 
given. 

This problem is given as a rule iu the Common School 
Arithmetics without a demonstration, but the pupil will 
find a demonstration in geometry or trigonometry which 
he should work out. After solving it, he will then under- 
stand why it is not given in the arithmetics. As a general 
thing, the student of mathematics should solve this problem 
every time he feels himself in doubt about it. 

Proportionality and Similarity. 

Definitions are to be handled as sharp cutting instru- 
ments. They are the tools of thought in mathematical 
studies. The terms of frequent occurrence in this chapter 
to be learned are: 1. Proportionality: '2. Similarity: 3. In- 
ternal Segments: 4. External Segments: 5. Similar Poly- 
gons; 6. Homologous Points, Lines, or Angles; ?. Ratio of 
Similitude: 8. Mean Proportional ; 9. Third Proportional; 
10. Fourth Proportional; 11. Extreme and Mean Ratio; 
12. Harmonically; 13. Harmonic Points. 

The essential propositions to be demonstrated are: 

1. A parallel to one side of a triangle, intersecting the 
other two sides, divides these sides proportionally. 

What relation exists between the sides and their internal 
segments? If the sides be produced, what relation exists 
between the sides and their external semgents? 

If two lines are intersected by any number of parallels, 
how are the two lines divided ? If a straight line divides 
two sides of a triangle into proportional internal segments, 
is the straight line parallel to the third side? Why? If 
proportional to the external segments? Why? 

2. In any triangle the bisector of an angle divides the op- 
posite side into segments proportional to the adjacent sides. 

If the angle be exterior, what modification must be 



240 GEOMETRY. 

made? Draw a figure and demonstrate. Give numerical 
values to the three sides of each triangle, and then find 
both the internal and the external segments. Is it necessary 
to know the angle in degrees in order to find the segments. 
Illustrate your answer. Give the converse of these propo- 
sitions, and demonstrate the same. 



Conditions of Similarity. 

3. Triangles are similar when they are mutually equian- 
gular. 

4. Triangles are similar when an angle in the one is equal 
to an angle in the other, and the sides including these angles 
are proportional. 

5. Triangles are similar when their homologous sides are 
proportional. 

6. Triangles having their sides respectively parallel or 
perpendicular are similar. 

The conditions of equality and similarity of triangles 
should now be compared in every respect, and the distinc- 
tions also be made between each of these and the equiva- 
lency of two triangles. 

7. The areas of two similar triangles are to each other as 
the squares of their homologous sides. 

The more general proposition, however, is, that the areas 
of two similar polygons are to each other as the squares 
of any two homologous lines. Frequently in the solution 
of a numerical or algebraic problem it becomes necessary 
to find a homologous line in one polygon when the two 
areas and the homologous line of the other polygon are 
given. In that case the homologous sides of two similar 
polygons have the same ratio as the square roots of their 
areas. Many arithmetical problems are solved by this 
principle. 



CONDITIONS OF SIMILARITY. 241 

Illustrative Exercises. 

1. A field is 160 by 60 rods: what will be the length and 
breadth of a tract of land 4 times as large ? 

2. The homologous sides of two similar triangles are to 
each other as 6 to 10, and the sum of their areas is 832 
square inches: find the area of each triangle. 

8. When two chords intersect within a circumference, their 
segments are reciprocally proportional. 

This proposition is more generally expressed thus: If 
through a fixed point within a circumference a chord is 
drawn, the product of the two segments is a constant in 
whatever direction the chord may be drawn. 

9. When two secants intersect within a circumference, the 
whole secants and their external segments are reciprocally 
proportional. 

The two triangles formed are mutually equiangular and 
similar. If one of the secants becomes a tangent, then the 
tangent is a mean proportional between the whole secant 
and its external segment. Also if a secant is drawn 
through a fixed point within a circumference, the product 
of the whole secant by its exterior segment is a constant 
in whatever direction the secant may be drawn. 

Propositions 6 and 7 are two of the most important in 
elementary geometry, and should be thoroughly mastered 
by the pupil. 

Problems in Equivalent Areas. 

1. To construct a square equivalent to two given squares. 
This problem may be extended so as to embrace any num- 
ber of squares. Let the pupil discover the method. 

2. To construct a square equivalent to the difference of 
two given squares. 

3. To construct a square equivalent to a given parallelo- 
gram, 



242 GEOMETRY. 

4. Upon a given straight line, to construct a rectangle 
equivalent to a given rectangle. 

5. To construct a triangle equivalent to a given polygon. 

6. To construct a rectangle equivalent to a given square, 
having the sum of its base and altitude equal to a given 
line. 

Three or four recitations should be given to these prob- 
lems, or at least as much as is necessary for the class to 
master them. All are important. 

Other problems following directly after these in treatises 
on geometry should next be mastered in detail. 

Regular Polygons. Measurement of the Circle. 

Definitions to be Learned. 

1. A Regular Polygon. Give examples. 

2. Center of a Regular Polygon. When Inscribed, when 
Circumscribed. What do these two words mean ? From 
what are they derived ? To what conjugation does the 
word scribo belong ? 

3. Perimeter of a Regular Polygon. Analyze the word 
" perimeter." From what language is it derived ? 

4. Define Apothem. What does it mean in Geometry ? 
Similar Arcs ? Similar Sectors ? 

5. Radius of a Regular Polygon. 

The discussion of Regular Polygons leads ultimately to 
the finding of the circumference of the circle. The steps 
are gradual, beginning with the equilateral triangle in- 
scribed in a circle, then a square, and so on to a regular 
polygon of a very great number of sides. In dealing with 
these geometrical figures, several things have to be kept 
constantly in mind: the number of sides of the polygon, 
the angles at the centre and how obtained, the ratio be- 
tween the radius or diameter and the perimeter of the poly- 
gon, its area, etc. 



REGULAR POLYGONS. 243 

The principal propositions of Elementary Geometry 
touching polygons will now be stated. 

1. If a circumference be divided into an equal number of 
arcs, and a chord be drawn in each arc, these chords will 
form the sides of a regular polygon. 

This proposition depends upon the equal divisions of the 
circumference. 

2. A circle may be circumscribed about, or inscribed 
within, any regular polygon. 

It is here assumed that these things can be done, but 
the learner should show how they are done. 

3. Regular polygons of the same number of sides are sim- 
ilar. Show that they are composed of similar triangles. 

4. The perimeters of two similar regular polygons are 
to each other as the radii of their circumscribed or of their 
inscribed circles ; and their areas are to each other as the 
squares of their radii, or of any two homologous lines. 

Of what special proposition is this a more general state- 
ment ? Is there a more general statement yet ? Why ? 

5. The area of a regular polygon is equal to one half the 
product of its perimeter by its apothem ; or its perimeter by 
half its apothem. 

Upon what proposition does this depend ? This propo- 
sition is stated in another form, namely, that the area of a 
regular polygon is equal to its perimeter multiplied by 
half the radius of the inscribed circle. 

To be done. 

At this stage in the learner's progress, he should — 

1. Inscribe a square in a given circle. 

2. Inscribe a regular hexagon in a given circle. 

3. Inscribe an equilateral triangle in a given circle. 

4. Inscribe a regular decagon in a given circle. 

5. Inscribe a regular polygon of fifteen sides. 



244 GEOMETRY. 

The 4th and 5th are very pretty problems which should 
be constructed without fail. 

To divide a circle into 6, 12, 24, 48, etc., equal parts is 

another but simpler phase of inscribing a certain class of 
regular polygons in a circle; also a circle can be easily di- 
vided into 5, 10, 20, etc., parts. 

Approximating the Circumference of a Circle. 

With, what has preceded, it is now necessary for the 
learner to make still further advances toward determining 
the circumference of a circle. 

It is easily shown that an arc of a circle is less than any 
line which envelops it and has the same extremities. As a 
corollary to this, the circumference of a circle must be less 
than the perimeter of any polygon circumscribed about it. 
So far two steps have been taken which the learner must 
carefully note. 

Again, if the number of sides of a regular polygon in- 
scribed in a circle be increased indefinitely, the apothem of 
the polygon will approach to the radius of the circle as its 
limit. Finally, the circumference of the circle is the limit 
to which the perimeters of regular inscribed and circum- 
scribed polygons approach when the number of their sides 
is increased indefinitely; consequently the area of the cir- 
cle is the limit of the areas of these polygons. 

The learner should not be content, however, with what has 
been thus briefly sketched. He should now proceed to find 
the length of the circle itself in terms of aright line. This 
is called the rectification of the curve. 

The first approximation is made under the following con- 
ditions : Given the perimeters of a regular inscribed and 
a similar circumscribed polygon, to find the perimeters of 
the regular inscribed and similar circumscribed polygons of 
double the number of sides. 



APPROXIMATING CIRCUMFERENCE OF A CIRCLE. 245 

Using the common notation, the two equations 

(1) y= y^-p an d 

(2) P' =? p are obtained. 

Here^? and P represent the perimeters of a regular in- 
scribed and regular circumscribed polygon of the same num- 
ber of sides, and p' and P' the perimeters of the regular in- 
scribed and circumscribed polygons of double the number 
of sides. From equations (1) and (2) the learner should 
compute some of the values given in the tables of approxi- 
mation. 

Instead of using the perimeters of the regular inscribed 
and circumscribed polygons, the radius and apotliem of a 
regular polygon may be employed to compute the radius and 
apotliem of the isoperimetric polygon of double the number 
of sides. 

Since the ratio of the circumference of a circle to its 
diameter is constant, this constant is represented by 7t, so 
that for any circle whose diameter is 2 i?, and circumfer- 
ence C 9 we have the relation 


-— = n y or = 27tB. 

Now if the circle be conceived as made up of an infinite 
number of triangles, all having their vertices at the center 
of the circle, then the circumference of the circle will form 
the sum of all the bases of these triangles, and the radius 
of the circle will be the altitude of these triangles ; that is, 
the difference between the sides of these isosceles triangles 
and their altitudes is infinitely small. Hence the area of 
the circle will evidently be 

73 

27tE X ~ = 7tK 2 = Area. 



246 GEOMETRY. 

Exercises on Polygons. 

There are certain relations existing among the sides, 
radius, apothem, and area of the regular inscribed polygons 
which the learner should determine by a combination of his 
algebraic and geometrical knowledge. 

Let the learner prove the following, where R = radius of 
the regular inscribed polygon, r = apothem, a = one side, 
A — anterior angle, c = angle at center: 

1. In a regular inscribed triangle, 2 = R V 3, r — \R, 
A = 60°, C = 120°. 

2. In an inscribed square, 2 = R V%, r = \R V2, A = 90°, 
C = 90°. 

3. In a regular inscribed hexagon, a = R, r = \R V3, 

A = 120°, G = 60°. 

1/5 1 

4. In a regular inscribed decagon, a = R 

r = iR ^10 + 2 Vt, A = 144°, c =36°. 

These exercises may very properly be extended so as to 
include the octagon, dodecagon, and other regular polygons. 

Maxima and Minima of Plane Figures. 

Definitions. 

1. Geometrical forms, under restricted conditions, may 
have a Maximum or minimum form. 

2. A Maximum Figure is the greatest one of its kind. 

3. A Minimum Figure is the least one of its kind. 

4. Isoperimetrical Figures are those which have equal pe- 
rimeters. 

The subject of maxima and minima is very briefly 
treated in American works. Owing to its extensive ap- 
plication not only in the solution of pure geometrical prob- 
lems, but in the calculus also, it deserves much more than a 



MAXIMA AND MINIMA OF PLANE FIGURES. 247 

mere passing notice. It was cultivated by the ancient 
geometers with marked success, and more recently by the 
English with much greater diligence than by those of the 
Continent. 

A few of the simple propositions only will be referred to 
in this connection, — enough, however, to give the learner a 
glimpse of this interesting subject. 

Propositions. 

1. The shortest distance from a given point outside of a 
given line is the perpendicular to the line. 

2. The greatest straight line that can he drawn in a circle 
is a diameter. 

3. The shortest straight line that can he drawn through a 
fixed point within a circumference is the chord at right 
angles to the line joining the center and the fixed point. 

4. The sum of the lines drawn from two fixed points on 
the same side of a given line to a point in that line is the 
least when the lines make equal angles with the fixed line. 

This is one of the most beautiful theorems in geometry. 

5. Of all triangles formed with two given sides, the 
greatest is that when these two given sides make a right 
angle. 

6. Of all triangles having the same hase and equivalent 
areas, the isosceles triangle has the least perimeter. 

7. The sum of two adjacent sides of a rectangle being 
constant, the area is the greatest when the sides are equal. 

8. Among isoperimetric triangles with a constant hase, 
the greatest is isosceles. 

How is the proposition modified if the base is not constant? 
Why? 

9. Of all plane figures containing the same area, the circle 
has the least perimeter. 

10. Of all equivalent polygons of the same number of 
sides, the one of the least perimeter is regular. 



248 GEOMETRY. 

11. Of all isoperiihetrical polygons of the same number of 
sides, the one which is regular has the greatest area. 

Exercises. 

1. With 360 panels of fence how can a farmer enclose 
the largest amount of ground ? Illustrate your answer. 

2. Divide the number 36 into such parts that their prod- 
ucts will be the greatest possible. What are the parts ? 

3. What three plane figures only can be used to cover 
completely the space about a point in a plane ? Why ? 

4. Find a point on a semi-circumference that the sum of 
the distances to the ends of diameter is the greatest possi- 
ble. 

5. How should the roof of a barn made of two slant sides 
be pitched so as to hold the most hay ? 

Geometry of Space. 

Plane Geometry is restricted to points, lines, angles, and 
surfaces lying in a plane. It is comprised within two 
dimensions, length and breadth. Geometry of Space in- 
cludes an additional dimension — thickness. A clear and 
comprehensive knowledge of plane figures lays the foun- 
dation for a correct understanding of Spacial Geometry. 
The geometry of the point is position ; of the line, extent; 
of the plane figure, length and breadth ; and of the solid, 
length, breadth, and thickness. 

All plane figures can be represented by suitable diagrams 
on a sheet of paper, slate, or blackboard ; but in Geometry 
of Space this is impossible. Whenever the learner fails to 
see mentally the figure he wishes to discuss, he should use 
sticks, wires, threads, or pins to represent lines, and stiff 
card-board or heavy pieces of paper to represent surfaces 
and solids. 






GEOMETRY OF SPACE —DEFINITIONS, ETC. 249 

In general, it is better to use these or similar helps in the 
demonstration of all difficult propositions. 

Definitions and General Principles. 

The learner must now familiarize himself with the mental 
process of passing from a plane into space, and to see spa- 
cial relations with as great facility as he has hitherto studied 
them in a plane. Here, however, he is obliged to depend 
upon definitions and general principles in order to make 
the most rapid progress. The following terms and princi- 
ples in connection with the axioms and postulates consti- 
tute the foundation of Geometry of Space, and so much 
importance is attached to them that progress is impossible 
until they are clearly and critically apprehended. 

To be Mastered. 

1. Plane. 2. When is a plane determined ? 3. How is 
a plane determined ? First? Second? Third? Fourth? 
Illustrate how a plane is determined by a line and a point. 
How is it not determined by a point and a line ? Illustrate 
how three points may or may not determine a plane. Under 
what condition will two straight lines determine a plane ? 
Will two straight lines that do not intersect determine a 
plane ? Illustrate. 4. When is a straight line perpendicular 
to a plane? 5. When oblique to a plane ? 6. When is a 
line parallel to a plane ? 7. When are two planes parallel ? 
8. AVhat is the projection of a point on a plane ? 9. What 
is the projection of a line on a plane? 10. Why will 
a straight line not determine the position of a plane? 
11. What is the inclination of a line to a plane ? 

Elementary Propositions to be Proved. 

1. The intersection of two planes is a straight line. 
Why? 



250 GEOMETRY, 

2. But one perpendicular can be drawn from a point out 
of a plane to that plane. 

3. All perpendiculars drawn to a given point in a line lie 
in one plane. 

4. A straight line perpendicular to two other straight 
lines at their point of intersection is also perpendicular to 
the plane in which the two straight lines lie. 

5. If from a point out of a plane lines be drawn meeting 
the plane at equal distances from the foot of the perpen- 
dicular, these lines are equal, and lines meeting the plane 
at the greater distance from this foot are longer. 

6. A straight line parallel to a line in a plane is also 
parallel to the plane. 

7. If a line is perpendicular to a plane, every line parallel 
to the given line is perpendicular to the plane. 

8. Two planes perpendicular to the same line are par 
allel. 

9. When two angles not in the same plane have their 
sides parallel and extend in tHe same direction, the angles 
are equal and their planes are parallel. 

10. If two straight lines are cut by three or more parallel 
planes, the corresponding segments are in proportion. 

Diedral Angles.— Angle of a Line and Plane, 

Definitions. 

1. When two planes meet and are terminated by their 
common intersection, they form a diedral angle. The planes 
are its faces, and the intersection is its edge. An open 
book represents a diedral angle. The two pages are the 
planes; and where the leaves join is their intersection. A 
diedral angle is measured by the difference of the opening 
between the two leaves, or by a straight line in each face 
drawn perpendicularly to the same point in the edge. An 
angle thus formed is called the plane angle of the diedral. 



DIEDRAL ANGLES. 251 

2. The magnitude of a diedral depends upon the differ- 
ence of the divergence between its two faces, i.e. from 0° 
to 360°. When the two faces form one plane, the angle is 
0°, or 180°, or 360°. They form two angles, an inside angle 
and an outside angle. Their sum is constant, and equals 
360° when one complete revolution is made. Hence a 
diedral angle may be any angular quantity. A diedral 
angle may be equal to any sum or to any difference of 
angles, or it may bear any ratio to any other angle. 

3. Diedrals are acute, obtuse, supplementary, etc. 

4. All right diedrals are equal. The learner will observe 
the agreements in 3 and 4 with those in regard to plane 
angles formed by intersecting straight lines. 

5. Wlien two planes pierce each other, the opposite or 
vertical angles are equal. How does this correspond to a 
similar proposition in Plane Geometry ? 

Propositions. 

1. Two diedral angles have the same ratio as their plane 
angles. 

2. When a straight line is perpendicular to a plane, every 
plane which passes through the line will he perpendicular 
to that plane. Why ? Illustrate. 

3. When two planes are perpendicular to each other, if a 
straight line in one is perpendicular to their intersection, 
it will he perpendicular to the other plane also. 

4. Through any straight line a plane can he passed per- 
pendicular to any given plane. 

5. If two intersecting planes are each perpendicular to 
a third plane, their intersection is also perpendicular to 
that plane. 

6. Every point of a plane which hisects a diedral angle 
is equally distant from its two faces. The hisecting plane 
is the locus of all the points equally distant from the two 



252 m GEOMETRY. 

given planes. How is this statement modified if the bisect- 
ing plane pierces the other two ? 

7. The projection of a point upon a plane is the foot of 
the perpendicular let fall from the given point upon the 
plane. Consequently, the projection of any line on a plane 
is the projections of all the points of the line upon the 
plane. 

8. The acute angle that a straight line makes with its 
own projection upon a plane is the least angle which it 
makes with any line of the plane. This angle is called the 
inclination of the line to the plane, or the angle formed by 
the line and the plane. 

Polyedral Angles. 

1. When three or more planes meet in a common point, they 
form a polyedral angle. The point where they meet is the 
vertex. The intersections of the planes are the edges, the 
portions of the planes between the edges are the faces, and 
the angles formed by the edges are face-angles. 

2. The simplest form of the polyedral is the triedral angle 
having three faces. 

3. The magnitude of a polyedral angle depends solely 
upon the relative position of the faces. 

4. Two polyedrals are equal when they can be made to 
coincide, or can be imagined as coinciding. 

5. A polyedral is convex when a plane cutting all its 
faces forms a convex polygon. 

6. Symmetrical polyedrals are those whose elements are 
respectively equal, but are arranged in a reverse order. If 
the edges of a polyedral be produced beyond the vertex, 
they will form the edges of § new polyedral which is sym- . 
metrical with the first. These two polyedrals are equal by 
symmetry, but they cannot be placed so that their faces will 
coincide. 






POLYEDBAL ANGLES. 253 

Remark. The learner will obtain a good idea of sym- 
metry by taking three or more pieces of card-board and 
putting them together so that they form symmetrical 
polydedral angles. A material illustration will greatly 
assist him in understanding thoroughly this special depart- 
ment of Geometry. 

Propositions. 

1. The sum of any two face-angles of a triedral angle is 
greater than the third face-angle. If the sum of the two 

face-angles become equal to the third face-angle, what will 
the triedral be ? Why ? 

2. The sum of the face-angles of any convex polyedral angle 
is less than four right angles. 

The usual demonstration given of this proposition is not 
always satisfactory to the learner. He sees the truthful- 
ness of the proposition, but the proof is slightly obscure. 

To make it plain at the beginning, take a point above 
the plane of the paper and imagine three lines drawn from 
the given point to three other points on the paper. These 
three lines will be the three edges of a triedral angle. Xow, 
let the vertex of this triedral be depressed towards the 
plane of the paper, and the limit of the face-angles will be 
360°. When the triedral becomes a plane triangle this 
limit is reached. Again, suppose the vertex to be raised 
indefinitely above the plane of the paper, and at the mo- 
ment when the edges of the triedral are parallel each ver- 
tical angle is 0°. Hence the limits are 0° and 360.° 

The learner is ready now to study with profit the usual 
demonstration. In the demonstration it is better to use 
all the inequations, rather than to use one or two and then 
infer the others, and finally jump at the conclusion. 

3. Two triedral angles are equal or symmetrical if the 
three face-angles of one are respectively equal to the three 
face-angles of the others. 



254 GEOMETRY. 

4. In every triedral the sum of the diedrals is greater than 
two right angles and less than six right angles. 

In this proposition the supplementary triedral may be 
considered in connection with the triedral; or better still, 
a pasteboard triedral can be used, and from a point within 
it let perpendiculars be drawn to each face; then the con- 
ditions expressed in the proposition can be easily and sim- 
ply demonstrated. 

The importance of thorough work here cannot be over- 
estimated, on account of its bearings on spherical trigo- 
nometry. The roots of that special department reach 
down deep into this proposition. 

Exercises. 

1. The three planes which bisect the diedral angles of a 
triedral meet in the same straight line. 

2. What is the locus of all the points at a given distance 
from a given plane ? 

3. Find the locus of all the points any one of which is 
equally distant from three given points. 

4. What is the locus of all the points any one of which 
is equally distant from three given planes ? 

5. If two convex polyedrals have the same number of 
faces, the sum of the diedrals of one is within four right 
angles of the sum of the diedrals of the other. 

6. If perpendiculars are drawn from a point within a 
diedral angle perpendicular to its faces, the angle included 
between the perpendiculars is the supplement of the diedral 
angle. 

7. If three perpendiculars are respectively drawn from a 
point within to the faces of a triedral, what are the inferior or 
superior limits of the three supplementary diedral angles ? 

8. Find the locus of the points any one of which is 
equally distant from two given points, and also equally dis- 






POLYEDRONS. 255 

tant from two given straight lines which lie in the same 
plane. 

9. Find the locus of the points which are equally distant 
from two given points, and also from two given lines. 

10. Find the locus of the points which are equally dis- 
tant from two given straight lines in the same plane, and 
also from two given planes. 

11. In how many different ways may three different 
planes intersect ? Illustrate by diagrams. 

Polyedrons. 

The study of diedrals and triedrals prepares the way for 
that of Polyedrons. Although the little child studies many 
of the geometrical forms in the lower grades, yet his 
knowledge of them at best is very inadequate compared to 
the scientific accuracy which is now demanded. But here 
the learner needs material forms to aid him in his investi- 
gations. 

Before he can see pure geometrical forms, he must obtain 
clear and exact ideas from the material illustrations repre 
senting the pure forms. Let him, therefore, construct each 
form as he learns a definition of it. The appeal made 
through sight and touch greatly re-enforce the imagina- 
tion in constructing an adequate conception of the form 
itself. Ideas obtained in this way are retained longer than 
those fleeting impressions which have no fixed abode in the 
mind. Here, again, knowledge rests on definitions, gener- 
alized and comprehended. 

The following terms should be defined, learned, and 
illustrated: 1. Polyedrons; 2. Faces; 3. Edges; 4. Ver- 
tices; 5. Diagonal; 6. Tetraedron ; 7. Hexaedron; 8. Oc- 
taedron ; 9. Dodecaedron ; 10. Icosaedron; 11. Convex; 
12. Volume ; 13. Measure ; 14. Unit ; 15. Equivalent ; 
16. Equal ; 17. Prism; 18. Surface; 19. Altitude; 20. Tri- 
angular; 21. Quadrangular; 22. Right; 23. Oblique; 



256 GEOMETRY. 

24. Regular; 25. Parallelopiped; 26. Cube; 27. Pyramid; 
28. Slant; 29. Truncated; 30. Frustum. 

Propositions. 

1. Parallel sections of a prism are equal polygons. 

2. The lateral surface of a right prism is equal to the 
perimeter multiplied by the base. 

8. The four diagonals of a parallelopiped bisect each other. 

4. The sum of the squares of the four diagonals of a paral- 
lelopiped is equal to the sum of the squares of its twelve 
edges. 

5. Two prisms are equal if three faces, including a triedral 
angle of the one, are respectively equal to three faces simi- 
larly arranged, including a triedral of the other. 

6. An oblique prism is equivalent to a right prism whose 
bases are equal to right sections of the oblique prism, and 
whose altitude is equal to a lateral edge of the oblique 
prism. 

7. Two rectangular parallelopipeds having equal bases are 
to each other as their altitudes. 

8. Two rectangular parallelopipeds are to each other as 
the products of their three dimensions. 

9. The volume of a rectangular parallelopiped is equal to 
the product of its length, breadth, and height. 

10. The volume of any parallelopiped is equal to the pro- 
duct of its base by its altitude ; also, the volume of any prism 
is equal to the product of its base by its altitude. 

11. When a pyramid is cut by a plane parallel to its base, 
the lateral edges and the altitude are divided proportionally, 
and the section is a polygon similar to the base. 

12. Two triangular pyramids having equal altitudes and 
equivalent bases are equivalent. 

13. The lateral area of a regular pyramid is equal to the 
perimeter of its base multiplied by one-half its slant height. 






POLYEDRONS. 257 

14. A triangular pyramid is one-third of a triangular 
prism of the same base and altitude. 

Remark. The learner should take a potato or an apple, 
and verify the 14th proposition by cutting a triangular prism 
from it, and cutting the prism into three equivalent tri- 
angular pyramids. The importance of this proposition 
cannot be overestimated, since it is the basis for ascertain- 
ing the volume of the cone and sphere as well as that of 
the pyramid. 

15. The volume of any pyramid is equal to one-third of 
the product of its base by its altitude. 

The volume of a frustum of a pyramid is equal to one-third 
of the product of its altitude by the sum of its lower base, its 
upper base, and the mean proportions between its bases. 

This is most readily demonstrated by completing the 
pyramid, and then finding the difference between the whole 
pyramid and the added one. 

17. Similar polyedrons are to each other as the cubes of 
their homologous edges. This proposition may be stated 
more generally thus: Two similar polyedrons are to each 
other as the cubes of any two homologous lines. 

Exercises. 

1. There is a point equally distant from the four vertices 
of a tetraedron. 

2. There is a point equally distant from the four faces of 
a tetraedron. 

3. The six planes that bisect the diedrals of a tetraedron 
meet in a point. 

4. A pyramid can be divided into the same number of 
tetraedrons as its base can be divided into triangles. 

5. The four straight lines that join the vertices of a 
tetraedron with the intersections of the medians of the 
opposite faces meet at a common point, and are in the ratio 
of 3 to 1. 



258 GEOMETRY. 

6. The distance of the center of a parallelopiped from 
any plane is equal to one-eighth of the sum of the distances 
of its eight vertices from the same plane. 

7. Cut a cube by a plane so that the section shall be a 
regular hexagon. 

8. The base of a regular pyramid is a hexagon each of 
whose sides is 3 meters in length, and its convex surface is 
6 times the area of its base : required its height. 

9. What is the volume of a rectangular parallelopiped 
whose length and width are as 3 to 2, and whose surface 
is 208 inches ? 

10. Prove that the base of a pyramid is less than its 
lateral surface. 

11. If the three dimensions of a rectangular parallelopiped 
be denoted by a, b, c, what would be the expressions for 
the surface, volume, and diagonals ? 

12. The edges of a regular tetraedron are 6 inches each: 
required the length of a straight line drawn from a vertex 
to the middle point of the opposite face. 

13. The entire surface of a regular tetraedron is to its 
volume as 4 to 5 : what is the length of one of its edges ? 

14. If a wheat-bin is 20 feet long, 12 feet wide, and 10 
feet deep, what will be the dimensions of a similar bin which 
holds ten times as much grain ? 

The Regular Polyedrons. 

This discussion depends upon the definition of the word 
Regular as it is applied to Polyedrons. It has been shown 
in Elementary Geometry that a plane may be covered about 
a point by six equilateral triangles, four squares, or by three 
regular hexagons, and that any other case is impossible. 

Consequently, the problem of the Kegular Polyedrons is 
to determine the number of regular convex polyedrons that 
can be constructed so as to enclose a definite form. 



THE REGULAR POLYEDRONS. 259 

To surround a limited space, a convex polyedral angle 
must have three faces, and the sum of the three face-angles 
must be less than 360°; for if the sum of the face-angles be 
360°, then the solid angle becomes a plane surface. 

1. By using one of the face-angles of an equilateral tri- 
angle as a measuring unit, we find that the following com- 
binations of convex polyedral angles may be formed, 
namely: of three, four, or five equilateral triangles. A 
fourth case is impossible, since six equilateral triangles will 
reduce the polyedral angle to a plane. 

2. It is also evident that a convex polyedral angle may 
be formed with three squares, but not with four. One case 
is therefore possible. 

3. Since a face-angle of a regular pentagon contains 
108°, a convex polyedral angle may be formed by com- 
bining three regular pentagons. Four such angles are 
greater than 360°, hence one case is possible. 

4. Three angles of a regular hexagon are equal to 360°, 
and of a regular heptagon greater than 360°; hence only 
five regular convex polyedrons are possible. They are 
named, from the number of faces composing each polyedral 
angle, as follows: 1. Tetraedron ; 2. Hexaedron, or Cube; 
3. jOctaedron ; 4. Dodecaedron : 5. Icosaedron. 

Before constructing these polyedrons, the learner should 
make models out of cardboard. They should be cut out 
entire, and the lines separating adjacent polygons should be 
cut half through the cardboard, and the edges turned to 
the proper form, when they can be sewed or glued. The 
tetraedron is composed of four regular equilateral triangles ; 
the cube, of six squares; the dodecaedron, of twelve regular 
pentagons; and the icosaedron, of twenty regular equilateral 
triangles. 

The next step is for the learner to construct each of 
these polyedrons, beginning with them in the order enumer- 
ated. To construct a regular polyedron, one of the faces, 



260 GEOMETBY. 

or an edge, should be known ; then all the other dimensions 
can be readily determined. 

Ruler's Theorem on Polyedrons. 

Leonard Euler, the distinguished mathematician, dis- 
covered the following beautiful theorem which bears his 
name: 

" In any polyedron, the number of its edges increased by 
two is equal to the number of its vertices increased by the 
number of its faces." 

The theorem may be expressed thus : E -\-2 = V -\- F, 
where E denotes the number of edges; V, the vertices; and 
F 9 the faces. 

This equation is easily demonstrated by taking any one 
of the five regular polyedrons and removing one face, and 
then noting the number of edges, of vertices, and of faces 
remaining. 

Beginning with one f&ce of an equilateral triangle, say, 
the edges equal the vertices; next annexing a second face, 
we have an edge and two vertices in common with the first. 

Or, for 2 faces, E = V+l; 
for 3 faces, F= V+2; 
for 4 faces, E = V -\- 3; and 
for N faces, E = V+ N- 1. 
F faces, E= V+F-2. 
Or, E+ 2 = V+F, 

which proves the theorem. 

Exercises. 

1. The sum of all the face-angles of any polyedron is 
equal to four right angles taken as many times, less two, as 
the polyedron has vertices. 

2. The polyedron which has for its vertices the centers 
of the four faces of a regular tetraedron is also a tetraedron 



THE THREE ROUND BODIES— THE CYLINDER. 261 

3. What ratio do the tetraedrons in No. 2 bear to each 
other,, if the edge of the circumscribing one be 10 inches ? 

4. An octaedron is inscribed in a cube, the vertices being 
at the centers of the faces of the cube: what part of the 
volume of the cube is the octaedron ? 

The Three Round Bodies. 

Elementar) r Geometry treats of only three bodies bounded 
by curved surfaces. These are the cylinder, the cone, 
and the sphere. It is more logical to treat each of these 
bodies in detail than to present a few of the properties of 
one and then begin with another, and so on. 

The Cylinder. 

Definitions. 

1. The cylinder. 2. How is it generated ? 3. Literal 
meaning. 4. Surface. 5. Bases. 6. Altitude. 7. Gener- 
atrix. 8. Directrix. 9. Eight cylinder. 10. Cylinder of 
Eevolution. 11. Eadius. 12. Circumscribed. 13. In- 
scribed. 11. Elements. 15. Volume. 16. A material 
cylinder. 17. Similar cylinders. 

Principles and Propositions. 

1. Every section of a cylinder embracing two elements is a 
parallelogram. Illustrate. 

2. Any section of a right cylinder parallel to its lower 
base is a circle. 

3. The convex surface of a cylinder is equal to the perim- 
eter of the base multiplied by its altitude. 

How must this statement be modified if the area of the 
two bases be included ? What formula represents the con- 
vex surface of a cylinder ? The entire surface ? 

4. The volume of a cylinder is equal to the product of 



262 GEOMETRY. 

its base by its altitude. What does V = nr* X a = nr* a 

mean ? 

The Cone. 

Definitions. 

1. Definition of Cone. 2. How generated. 3. The 
generatrix. 4. The directrix. 5. The vertex. 6. The 
base. 7. Altitude. 8. Surface. 9. Element. 10. Nap- 
pes. 11. Right. 12. Oblique. 13. Circular. 14. Vol- 
ume or Solidity. 15. Similar cones. 16. Truncated cone. 
17. Frustum. 18. Axis. 19. Slant. 20. Inscribed. 
21. Circumscribed. 

Properties. 

1. Any section made by passing a plane through the vertex 
of a cone and cutting its base is a triangle, 

2. If a plane cut a right cone parallel to its base, the sec- 
tion is a circle. 

3. The convex surface of a right cone is equal to the cir- 
cumference of its base multiplied by its slant height. 

What is the difference between the convex surface of a 
cone and its entire surface ? 

4. The convex surface of the frustum of a right cone is 
equal to the sum of its bases multiplied by half its slant 
height. 

5. The volume of a cone is equal to the area of its base 
multiplied by one third of its altitude. 

6. The volume of the frustum of a cone is equal to one 
third of the product of its altitude by the sum of its lower 
base, its upper base, and a mean proportional between the 
two bases. 

Exercises. 
1. The slant height of a right cone is twice the diameter 
of its base : what is the ratio of the area of the base to the 
lateral surface? 



THE SPHERE. 263 

2. How many yards of domestic f of a yard wide will be 
required to make a conical tent 15 feet in diameter and 
10 feet high ? 

3. Given the total surface 8 of a right cone, and the 
radius r of the base, to find the volume V. 

4. Given the total surface 8, and the lateral surface S' 9 
to find the volume V. 

5. The altitude of a right cone is 6 meters, and the di- 
ameter of its base 4 meters : required its slant height, lat- 
eral surface, area of base, and volume. 

6. The lateral area A of a right cone being given, what 
is the relation between its altitude a and the diameter D 
of its base ? 

7. Two similar cones are to each other as the cubes of 
their like dimensions. 

The Sphere. 

It would be more logical to treat of the surface and vol- 
ume of the sphere before passing to the consideration of 
spherical triangles and their properties. Yet this does not 
follow the usual order of presentation in most American 
treatises. 

Definitions. 

1. Sphere. 2. How generated. 3. Meaning of the word 
"sphere." 4. Diameter. 5. Eadius. 6. Center. 7. Sur- 
face. 8. Volume. 9. Great circle. 10. Small circle. 11. 
Tangent plane. 12. Point of contact. 13. Kegular semi- 
perimeter. 14. Eegular semi-polygon. 15. Inscribed cylin- 
der. 16. Circumscribed cone. 17. Inscribed cone. 18. 
Circumscribed cone. 19. Tangent spheres. 20. Concen- 
tric spheres. 



264 



GEOMETRY. 



Propositions. 

1. Any plane section of a sphere is a circle. 

2. The largest possible section is that which passes through 
the center of the sphere. 

3. Two great circles of a sphere bisect each other. 

4. The intersections of two circumferences of great circles 
on the surface of a sphere are distant 180°. 

5. The surface described by the revolution of a regular 
semi-polygon about its axis is equal to the product of the 
axis by the circumference of the inscribed circle. 

6. The area of the surface of a sphere is equal to the prod- 
uct of the diameter by the circumference of a great circle. 

7. The volume of a sphere is equal to one third of the 
product of the surface by the radius. 

8. The surface of a sphere is equal to the convex surface of 
the circumscribing cylinder. 

9. The volume of the sphere is two thirds of the circum- 
scribing cylinder. 

10. If a cone and a cylinder have for their respective bases a 
great circle of a sphere, and their altitudes are equal to the 
diameter of the sphere, their volumes are to each other as 
1:2:3. 



Spherical Surfaces and Volumes. 

The simplest form of spherical surfaces is the spherical 
angle. The geometrical figures that can be made on the 
surface of a sphere are as various as those in plane geom- 
etry. They are, however, as easily understood if properly 
approached. Two methods are adopted in presenting this 
subject: one begins with the figures on the surface of the 
sphere; the other conceives planes as intersecting at the 
center of the sphere and passing outward to the spherical 
surface which they cut. The latter method is the more 
satisfactory. The different kinds of spherical surfaces and 



SPHERICAL SURFACES AND VOLUMES. 265 

volumes may be grouped under the following heads, 
namely: 1. Spherical angles. 2. Spherical triangles. 3. 
Spherical polygons. 4. Line. 5. Wedge. 6. Zone. 7. 
Segments. 8. Sector. 9. Pyramid. 10. Hemisphere. In 
teaching Spherical Geometry the learner should be led to 
observe the analogies and contrasts between lines made on 
a plane and those made on the surface of a sphere. As 
illustrations: distances in plane geometry are measured 
along a straight line, while in spherical geometry distances 
are measured on the arc of a great circle. In plane geom- 
etry the sum of the three angles of any plane triangle is 
always constant — equal to 180°; but in spherical geometry 
the sum of the three angles is always greater than two 
right angles and less than six right angles. The first con- 
ception the learner obtains of a spherical angle should be 
clear and precise. Its measurement should be definitely 
fixed in his mind. To assist him, a blackboard sphere 
which revolves as easily as possible on its axis should be 
used to draw the triangles and their polars upon. Before 
demonstrating any of the properties of spherical triangles, 
the learner should devote some time to drawing triangles, 
their polars, and symmetrical triangles. An ordinary 
paper globe may be used advantageously in the absence of a 
regular blackboard globe. 

Propositions. 

1. A spherical angle is measured by the diedral angle 
which is included by the planes of its sides. 

2. Every spherical triangle corresponds to a triedral angle 
at the center of a sphere, having its six parts equal to the six 
parts of the spherical triangle. 

3. Any triedral angle may be represented by a triangle on 
the surface of a sphere. The vertex of the triedral is at the 
center of the sphere. 

Consequently : (1) If two sides of a spherical triangle are 



266 GEOMETRY. 

equal, the opposite angles are equal. (2) The greater side is 
opposite the greater angle, and conversely. (3) The sum of 
any two sides is greater than the third side. (4) Two 
spherical triangles on the same sphere are equal when 
their sides are equal and equally arranged. (5) The 
sum of the three sides of a spherical triangle is less 
than 360°. (6) Two opposite spherical triangles corre- 
spond to two opposite symmetrical triedral angles. (7) 
If two sides of a spherical triangle are quadrants, the 
angles opposite these sides are right angles. (8) Three 
great circles divide the surface of a sphere into eight 
spherical triangles, as three intersecting planes divide the 
space about a point into eight parts; hence every spherical 
triangle has seven others whose sides and angles are either 
equal or supplementary to those of the given triangle. 
(9) If the planes of three great circles are perpendicular to 
one another, they form eight equal tri-rectangular triedrals 
at the center of the sphere, and the eight corresponding 
triangles have all their angles right angles, and all their 
sides are quadrants. Such triangles are called quadrantal 
triangles. Lest the learner become confused in regard to 
spherical triangles and their triedrals, he needs to be 
cautioned that a spherical triangle has a definite area, and 
that its perimeter has a definite length, both depending 
upon the diameter of the sphere, while all the elements of 
a triedral are angular quantities and answer as well for a 
small sphere as a large one. 

4. The radii being equal, two spherical triangles are equal 
when — (1) The three sides are equal. (2) The three angles 
are equal. (3) Two sides and the included angle are equal. 
(4) One side and the adjacent angles are equal. (5) In all 
cases of equal elements, when the arrangement is the same, 
the triangles are equal; but when reversed, they are sym- 
metrical. 



POLAR TRIANGLES, 267 



Polar Triangles. 

If a perpendicular be erected to every face-angle at the 
vertex of a triedral angle, these lines form the edge of a 
supplementary triedral; and if the vertex be at the center 
of a sphere, two spherical triangles will be formed corre- 
sponding to the two supplementary triedrals. By construc- 
tion the edge of one triedral is perpendicular to the oppo- 
site face of the other, and the vertex of each angle of one 
triangle is the pole of the opposite side of the other. 

Triangles thus formed are called Polar Triangles, or 
Supplementary Triangles. They are otherwise defined 
thus: If from the vertices of a spherical triangle, as poles, 
arcs of great circles are described, a spherical triangle is 
formed, called the Polar Triangle of the first. 

1. When one triangle is the polar of another, the second is 
the polar triangle of the first, 

2. In two polar triangles each side of the one is the supple- 
ment of the opposite angle of the other. 

Measurement of Spherical Areas and Volumes. 
The methods of obtaining the surface and volume of a 
sphere have been given, and it now remains to find the sur- 
face and volume of parts of a sphere, such as lines, tri- 
angles, polygons, zone sections, and segments. The princi- 
ples pertaining to these several kinds of figures will now be 
given. 

1. The surface or volume of a lime is to the surface or vol- 
ume of the sphere as the angle of the lnne is to 360°. 

2. The area of a spherical triangle is equal to the excess of 
the sum of its angles over two right angles. 

3. The volume of a spherical triangle is to the volume of its 
sphere as the area of its base is to the surface of the sphere. 

4. The area of a spherical polygon is measured by the sum 
of its angles minus the product of two right angles multiplied 
by the number of sides of the polygon less two; and the vol- 



268 GEOMETRY. 

urne of any spherical polygon is to the volume of the sphere 
as the area of its base is to the surface of the sphere. 

5. The area of a zone is equal to the product of its altitude 
of the circumference of a great circle. 

6. The volume of a spherical sector is equal to the area of 
the zone that forms its base multiplied by one third of the 
radius of the sphere. 

7. The volume of a spherical segment is equal to the half 
sum of its bases multiplied by its altitude plus the volume of a 
sphere of which that altitude is the diameter. 

8. The volume of a spherical triangle, polygon, pyramid, or 
sector is equal to the area of its base multiplied by one 
third of the radius of the sphere. 

9. Two spheres are to each other as the cubes of their like 
dimensions. 

Exercises. 

1. The entire surface of a sphere is 36 square feet : what 
is the surface of a lune of this sphere, whose angle is 45° ? 

2. The angles of a spherical triangle are 60°, 70°, and 80° ; 
if the radius of the sphere be 10 feet, what is the area of 
the spherical triangle ? 

3. What would be the area of a spherical polygon on the 
same sphere as No. 2, provided its angles are 110°, 120°, 
150°, and 160° ? 

4. If the diameter of the earth be 7912 miles, find the 
surface of the torrid zone if its altitude be 3200 miles, and 
the earth is regarded as a sphere. 

5. In a sphere whose radius is r, find the height of a 
zone whose area is 27tr. 

6. A globe is 10 feet in diameter : how much of its sur- 
face could a squirrel see if placed 5 feet above it? 

7. The altitude and volume of a spherical segment of one 
base are given, to find the diameter of the sphere. 

8. The surface of a sphere can be completely covered 
with either 4, 8, or 20 equilateral spherical triangles. 



MODERN GEOMETRY. 269 

Demonstrating the Formulas. 

One of the most valuable exercises is to have the pupils 
demonstrate all the geometrical formulas, particularly those 
pertaining to the circle, pyramids, cylinder, cone, sphere, 
and parts of the sphere. If, at any time, the teacher dis- 
covers that a pupil is weakening in his knowledge on any 
particular point, give him a problem to solve that will 
cause him to go over the entire field and strengthen him- 
self. Knowledge, unless called into frequent use, slips 
away very easily ; but when it is once thoroughly anchored 
in the mind it is seldom or never entirely lost. 

Modern Geometry. 

The more recent developments in geometry are desig- 
nated Modern Geometry, as distinguished from the pure 
Euclidian Geometry. The progress has been so great, and 
the discoveries have been so wide and far-reaching, that the 
extension now constitutes a new science. As much as can be 
done here is to call attention to this department, and with 
the hope that all teachers and students of the Euclidian 
Geometry will give some attention, at least, to this subject. 
Many of the discoveries are very simple in character and 
may be very profitably discussed in elementary treatises. 
The subjects that may be thus introduced are : 1. Transver- 
sals. 2. Harmonic Proportion. 3. Anharmonic Ratio. 
4. Pole and Polar to the circle. 5. Reciprocal Polars. 
6. Radiccd Axes. 7. Centers of Similitude. 

Transversals. 

Definition. A Transversal is a straight line intersect- 
ing a system of straight lines. 

Definitions to oe learned. 1. Segments. 2. Adjacent 
Segments. 3. Non-adjacent Segments. 4. Complete Quad- 
rilateral. 



270 GEOMETRY. 

Propositions. 

1. If a transversal cuts the sides of a triangle or the 
sides produced, the product of three non-adjacent segments 
of the sides is equal to the product of the other three seg- 
ments. 

2. Conversely, if three points are taken on the sides of a 
triangle, or the side produced, so that the product of three 
lion-adjacent segments is equal to the product of the other 
three segments, then the three points are in the same 
straight line. 

This proposition is a simple test for ascertaining whether 
three points lie in the same straight line. 

3. Three straight lines drawn through the vertices of a 
triangle and any point in its plane divide the sides so that 
the product of three non-adjacent segments is equal to the 
product of the other three segments. 

4. Conversely, if three points are taken on the sides of 
a triangle, or on the sides produced, so that the product of 
three non-adjacent segments is equal to the product of the 
other three segments, the three straight lines from the op- 
posite vertices of the triangle meet in one point. 

Under the 4th proposition the following may be easily 
demonstrated: 

1. That the medians of a triangle meet in a point. 

2. That the bisectors of the angles of a triangle meet in 
a point. 

3. That the perpendiculars let fall from the vertices of 
a triangle meet in a point. 

Harmonic Proportion. 

Definitions. 
1. Harmonic Proportion is defined by an explanation, 
thus : Three quantities are in Harmonic Proportion when 
the first is to the third as the difference between the first 



ANHABMOmC BATIO. 271 

and second is to the difference between the second and 
third. 

2. A Pencil is a system of lines radiating from a point i n 
a plane. 

3. A Ray is one line of a pencil. 

4. The point from which the rays start is the Vertex. 

5. An Harmonic Pencil is one which cuts its transversals 
harmonically. 

6. The alternate rays of an harmonic pencil are called 
Conjugate. 

7. The four points in which a line is cut by an harmonic 
pencil are Harmonic Points. 

8. Two circles are said to intersect orthogonally when 
their tangents at the common point of intersection form a 
right angle. 

Propositions. 

1. If a line is divided harmonically, the distance be- 
tween the points of division is the harmonic mean between 
the distances from the ends of the line to the point of di- 
vision not between them. 

2. If a straight line AB is divided harmonically at the 
points and D, then half of AB is a mean proportional 
between the distances of its middle point from the points 
C and D. Or to prove the relation OB = OC X OD. 
The converse of this proposition is also true. 

3. All transversals of a harmonic pencil are divided har- 
monically at the points of intersection. 

Anharmonic Ratio. 

Definitions. 

1. The anharmonic ratio of four points in a straight line, 

in which A, B, 0, D represent the points in the order of 
their arrangement, may be expressed thus : A X BD -f- 
AD x BO. 

2. The anharmonic ratio of a pencil of four rays is the 



272 GEOMNTRY. 

anharmonic ratio of the four points of intersection on these 
rays as determined by any transversal. 

3. The anharmonic ratio of four fixed points in the cir- 
cumference of a circle is the anharmonic ratio of the pen- 
cil formed by joining these four points with any point in 
the circumference. 

4. The anharmonic ratio of four tangents is the anhar- 
monic ratio of the points of four tangents by a fifth tan- 
gent. 

Propositions. 

1. The anharmonic ratio of four points is not changed by 
changing two of the letters, provided the other two letters 
are changed at the same time. 

2. If a pencil of four rays be cut by a transversal, any 
harmonic ratio of the four points of intersection is con- 
stant. 

3. The anharmonic ratio of four fixed points in the cir- 
cumference of a circle is constant. 

4. The anharmonic ratio of four fixed tangents to a cir- 
cle is constant. 

5. When two pencils have the same anharmonic ratio 
and a common homologous ray, the intersections of the 
other three pairs of homologous rays are in the same 
straight line. 

6. If two straight lines have the same anharmonic ratio, 
and a homologous point common, the straight lines joining 
the other three pairs of homologous points meet in the same 
point. 

7. The intersections of the three pairs of opposite sides 
of an inscribed hexagon are in the same straight line. 

8. The three diagonals which join the opposite vertices 
of a circumscribed hexagon meet in the same point. 

Remark. Proposition 7th is known as Pascal's Theorem, 
and - the 8th as Brian chon's. They are two of the most 



POLE AND POLAR TO A CIRCLE 273 

beautiful theorems in geometry. From them are derived 
several other interesting propositions that will well repay 
the student for the time devoted to them. 

Pole and Polar to a Circle. 

Definitions. 

1. A point is the pole of a fixed line, and a line is the 
polar of a fixed point with reference to a circle, if the line 
and point are so located that the chord of a revolving se- 
cant through the point is divided harmonically at the fixed 
point and the intersection of the fixed line with the secant. 
Or, if from a fixed point P a line be drawn to C, the center 
of a circle, and on the line CP a point p be taken such 
that CP X Cp is equal to the square of the radius of the 
circle, then the straight line Dp perpendicular to the line 
Cp at p is the polar of P with respect to the circle, and 
the point P is the pole of Dp. 

2. The intersection of the polar with the diameter 
through the pole is the polar point. 

Propositions. 

1. Any straight line drawn through the pole to meet a 
circle is divided harmonically by the circle and the polar. 

2. The polar of a given point with respect to a circle 
is a straight line perpendicular to the diameter drawn 
through the given point. 

Corollaries. 

1. The radius of the circle is a mean proportional be- 
tween the distances of the pole and its polar from the cen- 
ter. 

2. If the pole is without, its polar is the chord joining 
the points of tangency of the tangents drawn to the pole. 

3. If the pole is on the circumference, its polar is the 
tangent at the pole. 



274 GEOMETRY. 

4. The pole and polar are interchangeable. 

5. The polars of all points of a straight line pass 
through the pole of that line; and the poles of all straight 
lines which pass through a fixed point are situated on the 
polar of that point. 

6. The pole of a straight line is the intersection of the 
polars of any two points. 

7. The polar of any point is the straight line joining the 
poles of any two straight lines through that point. 

Reciprocal Polars. 

Definitions. 

1. Reciprocal polars are two polygons so related that the 
vertices of either are respectively the poles of the sides of 
the other with respect to the same circle, which is called 
the auxiliary circle. 

2. A theorem inferred from another by means of recip- 
rocal polars is called a Reciprocal Theorem. 

As examples of reciprocal polars, Brianchon's Theorem 
may be inferred from Pascal's Theorem, or Pascal's in- 
ferred from Brianchon's. 

Propositions. 

1. The angle contained by two straight lines is equal to 
the angle contained by the straight lines joining their 
poles to the center of the auxiliary circle. 

2. The ratio of the distances of any two points from the 
center . of the auxiliary circle is equal to the ratio of the 
distances of each point from the pole of the other. 

Radical Axis. 

Definitions. 
1. The Power of a Point in the plane of a circle is the 
rectangle of the segments, external or internal, into which 
the point divides the chord passing through it. 



BADIGAL AXIS—CENTEBS OF SIMILITUDE. 275 

5. The Radical Axis of two circles is the locus of the 
point whose powers with respect to the circles are equal. 

Propositions. 

1. Find the locus of a point from which tangents drawn 
to two given circles are equal. 

This proposition demonstrates that the radical axis of 
two circles is a straight line perpendicular to the line join- 
ing the centers of the circles, and dividing this Hue so that 
the difference of the squares of the two segments is equal to 
the difference of the squares of the radii. 

Remark, If the two circles have no point in common, 
the radical axis does not intersect either of them; if the 
circles are tangent, externally or internally, their common 
tangent is their radical axis; and if they intersect, their 
common chord is their radical axis. 

2. The radical axes of a system of three circles, taken 
two and two, meet in a point. 

Centers of Similitude. 

Definitions. 
The External and Internal Centers of Similitude of 

two circles are two points which divide the line joining 
their centers harmonically in the ratio of the two radii. 

Propositions. 

1. If in two circles two parallel radii are drawn, one in 
each circle, the straight line joining their extremities in- 
tersects the line of centers in the external center of simili- 
tude if the parallel radii are in the same direction, and in 
the internal center of similitude if these radii are in oppo- 
site directions. 

2. The external centers of similitude of three circles, 
taken two and two, are in a straight line. 



276 GEOMETRY. 

The preceding definitions and propositions will enable 
the student to obtain some insight into this branch of 
Geometry. It has been touched but slightly in Ameri- 
can text-books, although many valuable contributions have 
appeared in the various mathematical magazines during 
recent years. 

Here is an excellent field for some American author to 
occupy. 

Conclusion. 

The teacher should select the exercises for his classes 
with great care. Step by step a class can be led to solve 
problems of great difficulty. By encouraging and not dis- 
couraging, the highest proficiency is attained. Never let a 
pupil lose confidence in himself. The genius of sticking 
to a problem till light begins to dawn is the chief element 
in mathematical skill. 

Suppose that a teacher wishes to lead his class up to 
such a problem as this: "To describe a circle that shall 
touch three given circles." He must start them at first with 
the simplest exercise, and lead upward gradually, thus: 

1. To describe a circle that shall pass through two given 
points and be tangent to a given straight line. 

2. To describe a circle that shall pass through two given 
points and be tangent to a given circle. 

3. To describe a circle that shall pass through a given 
point and be tangent to two given straight lines. 

4. To describe a circle that shall pass through a given 
point and shall touch a given straight line and a given 
circle. 

5. To describe a circle that shall touch a given circle 
and be tangent to a given line at a given point. 

6. To describe a circle that shall pass through a given 
point and be tangent to two given circles. 



CONCLUSION. 277 

7. To describe a circle that shall be tangent to two 
given circles and a given straight line. 

8. To describe a circle that shall be tangent to three given 
circles. 

If the learner begins with the first problem after he has 
had the necessary training, and solves it, discussing all 
possible cases, and so on with each of the others in detail, 
the 8th will present little difficulty. When he reaches the 
Modern Geometry, he will then discover another and very 
much simpler method of demonstration than the one out- 
lined in the Euclidian Geometry. 

De Quincey says: The act of learning a science is 
good, not only for the knowledge w r hich results, but for 
the exercise which attends it: the energies which the 
learner is obliged to put forth are true intellectual ener- 
gies; and his very errors are full of instruction. He fails 
to construct some leading idea, or he even misconstructs 
it : he places himself in a false position with respect to 
certain propositions; views them from a false center; 
makes a false or an imperfect antithesis; apprehends a 
definition with insufficient rigor, or fails in his use of it to 
keep it self-consistent. These and a thousand other errors 
are met by a thousand appropriate resources — all of a true 
intellectual character: comparing, combining, distinguish- 
ing, generalizing, subdividing, acts of abstraction and 
evolution, of synthesis and analysis, until the most tor- 
pid minds are ventilated and healthily excited by this 
introversion of the faculties upon themselves. 



